pdf2htmlEX/demo/cheat4.page

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"/><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Theoretical<span class="_ _0"> </span>Computer<span class="_ _0"> </span>Science<span class="_ _0"> </span>Cheat<span class="_ _0"> </span>Sheet</div><div class="t m0 x18e h3 y2 ff2 fs0 fc0 sc0 ls0 ws0">T<span class="_ _5"></span>rigonometry<span class="_ _4e"> </span>Matrices<span class="_ _4f"> </span>More T<span class="_ _5"></span>rig.</div><div class="t m0 x11e h3 y2f1 ff3 fs0 fc0 sc0 ls0 ws0">A</div><div class="t m0 xc8 h3 y2f2 ff3 fs0 fc0 sc0 ls0 ws0">c</div><div class="t m0 xd9 h3 y50 ff3 fs0 fc0 sc0 ls0 ws0">θ</div><div class="t m0 x107 h3 y2f3 ff3 fs0 fc0 sc0 ls0 ws0">B</div><div class="t m0 x6 h3 y2f4 ff3 fs0 fc0 sc0 ls0 ws0">a</div><div class="t m0 xc6 h3 y4f ff3 fs0 fc0 sc0 ls0 ws0">b</div><div class="t m0 x141 h3 y23c ff3 fs0 fc0 sc0 ls0 ws0">C</div><div class="t m0 x119 h8 y12b ffd fs3 fc0 sc0 ls0 ws0">(0,-1)</div><div class="t m0 x119 h8 y2f5 ffd fs3 fc0 sc0 ls0 ws0">(0,1)</div><div class="t m0 x10a h8 y129 ffd fs3 fc0 sc0 ls0 ws0">(-1,0)<span class="_ _50"> </span>(1,0)</div><div class="t m0 xc9 h3 y2f6 ff2 fs0 fc0 sc0 ls0 ws0">(cos<span class="_ _6"> </span><span class="ff3 ls10">θ,<span class="_ _6"> </span></span>sin<span class="_ _8"> </span><span class="ff3">θ</span>)</div><div class="t m0 x80 h3 y2f7 ff2 fs0 fc0 sc0 ls0 ws0">Pythagorean theorem:</div><div class="t m0 x109 h3 y2f8 ff3 fs0 fc0 sc0 ls0 ws0">C</div><div class="t m0 x148 h5 y279 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6 h3 y2f9 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">A</span></div><div class="t m0 x10b h5 y279 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xf5 h3 y2f9 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">B</span></div><div class="t m0 x132 h5 y279 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xd7 h3 y2f9 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 y2fa ff2 fs0 fc0 sc0 ls0 ws0">Definitions:</div><div class="t m0 xe2 h3 y252 ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">a<span class="_ _7"> </span></span>=<span class="_ _7"> </span><span class="ff3">A/C<span class="_ _3"></span>,<span class="_ _c"> </span></span>cos<span class="_ _8"> </span><span class="ff3">a<span class="_ _8"> </span></span>=<span class="_ _7"> </span><span class="ff3 ls15">B/<span class="_ _5"></span>C<span class="_ _5"></span>,</span></div><div class="t m0 xe2 h3 y2fb ff2 fs0 fc0 sc0 ls0 ws0">csc<span class="_ _6"> </span><span class="ff3">a<span class="_ _7"> </span></span>=<span class="_ _7"> </span><span class="ff3 ls15">C/<span class="_ _5"></span>A<span class="_ _3d"></span>,<span class="_ _c"> </span><span class="ff2 ls0">sec<span class="_ _6"> </span><span class="ff3">a<span class="_ _7"> </span></span>=<span class="_ _7"> </span></span>C/<span class="_ _5"></span>B,</span></div><div class="t m0 x92 h3 y38 ff2 fs0 fc0 sc0 ls0 ws0">tan<span class="_ _6"> </span><span class="ff3">a<span class="_ _7"> </span></span>=</div><div class="t m0 x18f h3 y257 ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">a</span></div><div class="t m0 x106 h3 y25a ff2 fs0 fc0 sc0 ls0 ws0">cos<span class="_ _6"> </span><span class="ff3">a</span></div><div class="t m0 x108 h3 y38 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x6 h3 y257 ff3 fs0 fc0 sc0 ls0 ws0">A</div><div class="t m0 x6 h3 y25a ff3 fs0 fc0 sc0 ls0 ws0">B</div><div class="t m0 xe6 h3 y38 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _c"> </span><span class="ff2">cot<span class="_ _6"> </span></span>a<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x190 h3 y257 ff2 fs0 fc0 sc0 ls0 ws0">cos<span class="_ _6"> </span><span class="ff3">a</span></div><div class="t m0 x190 h3 y25a ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">a</span></div><div class="t m0 x191 h3 y38 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xf7 h3 y257 ff3 fs0 fc0 sc0 ls0 ws0">B</div><div class="t m0 xf7 h3 y25a ff3 fs0 fc0 sc0 ls0 ws0">A</div><div class="t m0 xc4 h3 y38 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 y25f ff2 fs0 fc0 sc0 ls0 ws0">Area, radius of inscrib<span class="_ _3"></span>ed circle:</div><div class="t m0 x107 h5 y2fc ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x107 h5 y3b ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xf2 h3 y2fd ff3 fs0 fc0 sc0 ls0 ws0">AB<span class="_ _3"></span>,</div><div class="t m0 x82 h3 y62 ff3 fs0 fc0 sc0 ls0 ws0">AB</div><div class="t m0 xe6 h3 y2cb ff3 fs0 fc0 sc0 ls0 ws0">A<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>B<span class="_ _7"> </span><span class="ff2">+<span class="_ _8"> </span></span>C</div><div class="t m0 x10d h3 y2fe ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 y43 ff2 fs0 fc0 sc0 ls0 ws0">Iden<span class="_ _5"></span>tities:</div><div class="t m0 x80 h3 y1f1 ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=</div><div class="t m0 xc8 h3 y6a ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xc6 h3 y2ff ff2 fs0 fc0 sc0 ls0 ws0">csc<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x18f h3 y1f1 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _51"> </span><span class="ff2">cos<span class="_ _6"> </span></span>x<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x7 h3 y6a ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x192 h3 y2ff ff2 fs0 fc0 sc0 ls0 ws0">sec<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x15d h3 y1f1 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h3 y300 ff2 fs0 fc0 sc0 ls0 ws0">tan<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=</div><div class="t m0 x18c h3 y301 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x87 h3 y28a ff2 fs0 fc0 sc0 ls0 ws0">cot<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x143 h3 y302 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _52"> </span><span class="ff2">sin</span></div><div class="t m0 x13c h5 y6d ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x119 h3 y300 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span>cos</span></div><div class="t m0 x8e h5 y6d ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xca h3 y300 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _7"> </span><span class="ff2 ls1">=1<span class="_ _b"></span><span class="ff3 ls0">,</span></span></div><div class="t m0 x80 h3 y303 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span>+<span class="_ _8"> </span>tan</div><div class="t m0 x86 h5 y167 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x87 h3 y304 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span>sec</span></div><div class="t m0 xf2 h5 y167 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x109 h3 y304 ff3 fs0 fc0 sc0 ls0 ws0">x,<span class="_ _46"> </span><span class="ff2">1<span class="_ _8"> </span>+<span class="_ _8"> </span>cot</span></div><div class="t m0 x4 h5 y167 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1 h3 y304 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span>csc</span></div><div class="t m0 x136 h5 y167 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x179 h3 y304 ff3 fs0 fc0 sc0 ls0 ws0">x,</div><div class="t m0 x80 h3 y7d ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=<span class="_ _7"> </span>cos</div><div class="t m0 x12f h6 y1b3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x18b h5 y305 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x106 h5 y306 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x130 h4 y7d ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 xf3 h6 y1b3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x2 h4 y7d ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _41"> </span><span class="ff2">sin<span class="_ _6"> </span></span>x<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span>sin(</span>π<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>x<span class="ff2">)</span>,</div><div class="t m0 x80 h4 y307 ff2 fs0 fc0 sc0 ls0 ws0">cos<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=<span class="_ _7"> </span><span class="ff4">−<span class="_ _8"></span></span>cos(<span class="_ _5"></span><span class="ff3">π<span class="_ _7"> </span><span class="ff4">−<span class="_ _8"></span></span>x<span class="ff2">)</span>,<span class="_ _4a"> </span><span class="ff2">tan<span class="_ _8"> </span></span>x<span class="_ _8"> </span><span class="ff2">=<span class="_ _7"> </span>cot</span></span></div><div class="t m0 x176 h6 y16d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x8e h5 y1b8 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x8e h5 y308 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x192 h4 y309 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 x14f h6 y16d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x15d h3 y309 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h4 y30a ff2 fs0 fc0 sc0 ls0 ws0">cot<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=<span class="_ _7"> </span><span class="ff4">−<span class="_ _8"></span></span>cot(<span class="_ _5"></span><span class="ff3">π<span class="_ _7"> </span><span class="ff4">−<span class="_ _8"></span></span>x<span class="ff2">)</span>,<span class="_ _38"> </span><span class="ff2">csc<span class="_ _6"> </span></span>x<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span>cot</span></span></div><div class="t m0 xc3 h5 y26e ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 xc3 h5 y290 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x8e h4 y30b ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">cot<span class="_ _6"> </span><span class="ff3">x,</span></span></div><div class="t m0 x80 h4 y30c ff2 fs0 fc0 sc0 ls0 ws0">sin(<span class="ff3">x<span class="_ _8"> </span><span class="ff4">±<span class="_ _8"> </span></span>y<span class="_ _3"></span></span>)<span class="_ _7"> </span>=<span class="_ _7"> </span>sin<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>cos<span class="_ _6"> </span><span class="ff3">y<span class="_ _8"> </span><span class="ff4">±<span class="_ _8"> </span></span></span>cos<span class="_ _8"> </span><span class="ff3">x<span class="_ _2"> </span></span>sin<span class="_ _8"> </span><span class="ff3 ls10">y,</span></div><div class="t m0 x80 h4 y30d ff2 fs0 fc0 sc0 ls0 ws0">cos(<span class="ff3">x<span class="_ _8"> </span><span class="ff4">±<span class="_ _8"> </span></span>y<span class="_ _3"></span></span>)<span class="_ _7"> </span>=<span class="_ _7"> </span>cos<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>cos<span class="_ _6"> </span><span class="ff3">y<span class="_ _8"> </span><span class="ff4">∓<span class="_ _8"> </span></span></span>sin<span class="_ _8"> </span><span class="ff3">x<span class="_ _2"> </span></span>sin<span class="_ _8"> </span><span class="ff3 ls10">y,</span></div><div class="t m0 x80 h4 y30e ff2 fs0 fc0 sc0 ls0 ws0">tan(<span class="ff3">x<span class="_ _8"> </span><span class="ff4">±<span class="_ _8"> </span></span>y<span class="_ _3"></span></span><span class="ls1">)=</span></div><div class="t m0 x130 h4 y17b ff2 fs0 fc0 sc0 ls0 ws0">tan<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span><span class="ff4">±<span class="_ _8"> </span></span></span>tan<span class="_ _8"> </span><span class="ff3">y</span></div><div class="t m0 x147 h4 y30f ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">∓<span class="_ _8"> </span></span>tan<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>tan<span class="_ _2"> </span><span class="ff3">y</span></div><div class="t m0 x128 h3 y310 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h4 y2d5 ff2 fs0 fc0 sc0 ls0 ws0">cot(<span class="ff3">x<span class="_ _8"> </span><span class="ff4">±<span class="_ _8"> </span></span>y<span class="_ _3"></span></span><span class="ls1">)=</span></div><div class="t m0 x143 h4 y311 ff2 fs0 fc0 sc0 ls0 ws0">cot<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>cot<span class="_ _2"> </span><span class="ff3">y<span class="_ _7"> </span><span class="ff4">∓<span class="_ _8"> </span></span></span>1</div><div class="t m0 x130 h4 y312 ff2 fs0 fc0 sc0 ls0 ws0">cot<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span><span class="ff4">±<span class="_ _8"> </span></span></span>cot<span class="_ _8"> </span><span class="ff3">y</span></div><div class="t m0 x145 h3 y2d5 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h3 y2d8 ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span>2<span class="ff3">x<span class="_ _7"> </span></span>=<span class="_ _7"> </span>2<span class="_ _8"> </span>sin<span class="_ _2"> </span><span class="ff3">x<span class="_ _8"> </span></span>cos<span class="_ _2"> </span><span class="ff3">x,<span class="_ _49"> </span></span>sin<span class="_ _8"> </span>2<span class="_ _5"></span><span class="ff3">x<span class="_ _7"> </span><span class="ff2">=</span></span></div><div class="t m0 xc9 h3 ybe ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _6"> </span>tan<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 x1 h3 ybd ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span>+<span class="_ _8"> </span>tan</div><div class="t m0 x193 h5 y313 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x14f h3 ybd ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x15d h3 y2d8 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h3 y29b ff2 fs0 fc0 sc0 ls0 ws0">cos<span class="_ _6"> </span>2<span class="ff3">x<span class="_ _7"> </span></span>=<span class="_ _7"> </span>cos</div><div class="t m0 x100 h5 y314 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x143 h4 y29b ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">sin</span></span></div><div class="t m0 x142 h5 y315 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x94 h3 y29b ff3 fs0 fc0 sc0 ls0 ws0">x,<span class="_ _4"> </span><span class="ff2">cos<span class="_ _8"> </span>2<span class="_ _5"></span><span class="ff3">x<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span>2<span class="_ _6"> </span>cos</span></span></span></div><div class="t m0 xe7 h5 y314 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xca h4 y29b ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">1</span></span>,</div><div class="t m0 x80 h4 y2de ff2 fs0 fc0 sc0 ls0 ws0">cos<span class="_ _6"> </span>2<span class="ff3">x<span class="_ _7"> </span></span><span class="ls1">=1<span class="_ _5"></span><span class="ff4 ls0">−<span class="_ _8"> </span><span class="ff2">2<span class="_ _6"> </span>sin</span></span></span></div><div class="t m0 xe4 h5 y192 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x115 h3 y2de ff3 fs0 fc0 sc0 ls0 ws0">x,<span class="_ _53"> </span><span class="ff2">cos<span class="_ _6"> </span>2</span>x<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x1 h4 y316 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>tan</div><div class="t m0 x193 h5 y317 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x14f h3 y318 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x1 h3 y319 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span>+<span class="_ _8"> </span>tan</div><div class="t m0 x193 h5 y31a ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x14f h3 y319 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x15d h3 y2de ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h3 y2ed ff2 fs0 fc0 sc0 ls0 ws0">tan<span class="_ _6"> </span>2<span class="ff3">x<span class="_ _7"> </span></span>=</div><div class="t m0 x93 h3 y31b ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _6"> </span>tan<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 xc8 h4 y31c ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>tan</div><div class="t m0 x131 h5 y31d ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x18e h3 y31c ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x149 h3 y2ed ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _45"> </span><span class="ff2">cot<span class="_ _6"> </span>2</span>x<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x1 h3 y31e ff2 fs0 fc0 sc0 ls0 ws0">cot</div><div class="t m0 x8e h5 y31f ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x171 h4 y320 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">1</span></span></div><div class="t m0 x176 h3 y321 ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _6"> </span>cot<span class="_ _8"> </span><span class="ff3">x</span></div><div class="t m0 x15d h3 y2ed ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x80 h4 y2af ff2 fs0 fc0 sc0 ls0 ws0">sin(<span class="ff3">x<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">y<span class="_ _3"></span></span>)<span class="_ _6"> </span>sin(<span class="ff3">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>y<span class="_ _15"></span></span>)<span class="_ _7"> </span>=<span class="_ _7"> </span>sin</div><div class="t m0 x95 h5 y322 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x145 h4 y2af ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">sin</span></span></div><div class="t m0 x190 h5 y322 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x134 h3 y2af ff3 fs0 fc0 sc0 ls10 ws0">y,</div><div class="t m0 x80 h4 y323 ff2 fs0 fc0 sc0 ls0 ws0">cos(<span class="ff3">x<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">y<span class="_ _3"></span></span>)<span class="_ _6"> </span>cos(<span class="_ _3"></span><span class="ff3">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>y<span class="_ _3"></span></span>)<span class="_ _7"> </span>=<span class="_ _7"> </span>cos</div><div class="t m0 x82 h5 y324 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x128 h4 y325 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">sin</span></span></div><div class="t m0 x119 h5 y326 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xc2 h3 y325 ff3 fs0 fc0 sc0 ls10 ws0">y.</div><div class="t m0 x80 h3 y19d ff2 fs0 fc0 sc0 ls0 ws0">Euler’s equation:</div><div class="t m0 xc6 h3 y19e ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x194 h5 y2bc ff6 fs1 fc0 sc0 ls0 ws0">ix</div><div class="t m0 x12f h3 y19e ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>cos<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">i<span class="_ _8"> </span></span>sin<span class="_ _2"></span><span class="ff3">x,<span class="_ _4"> </span>e</span></div><div class="t m0 x13c h5 y2bc ff6 fs1 fc0 sc0 ls0 ws0">iπ</div><div class="t m0 x134 h4 y19e ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−</span>1<span class="ff3">.</span></div><div class="t m0 x153 h3 y327 ff2 fs0 fc0 sc0 ls0 ws0">Multiplication:</div><div class="t m0 xb3 h4 y11b ff3 fs0 fc0 sc0 ls0 ws0">C <span class="ff2">=<span class="_ _7"> </span></span>A<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span><span class="ls15">B,<span class="_ _c"> </span>c</span></div><div class="t m0 x138 h5 y328 ff6 fs1 fc0 sc0 ls0 ws0">i,j</div><div class="t m0 x14 h3 y11b ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x195 h5 yd ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x6b h6 y239 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x6b h5 y2f6 ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=1</span></div><div class="t m0 xce h3 y11b ff3 fs0 fc0 sc0 ls0 ws0">a</div><div class="t m0 x77 h5 y328 ff6 fs1 fc0 sc0 ls0 ws0">i,k</div><div class="t m0 x32 h3 y11b ff3 fs0 fc0 sc0 ls0 ws0">b</div><div class="t m0 x30 h5 y328 ff6 fs1 fc0 sc0 ls10 ws0">k,j</div><div class="t m0 x33 h3 y11b ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x153 h4 y329 ff2 fs0 fc0 sc0 ls0 ws0">Determinan<span class="_ _5"></span>ts:<span class="_ _1e"> </span>det<span class="_ _6"> </span><span class="ff3">A<span class="_ _7"> </span><span class="ff4"></span></span><span class="ls1">=0<span class="_ _3"></span>i<span class="_ _b"></span>ff<span class="_ _15"></span><span class="ff3 ls0">A <span class="ff2">is non-singular.</span></span></span></div><div class="t m0 x5b h4 y32a ff2 fs0 fc0 sc0 ls0 ws0">det<span class="_ _6"> </span><span class="ff3">A<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>B </span>=<span class="_ _7"> </span>det<span class="_ _8"> </span><span class="ff3">A<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"></span></span></span>det<span class="_ _2"></span><span class="ff3 ls15">B,</span></div><div class="t m0 x2b h3 y24a ff2 fs0 fc0 sc0 ls0 ws0">det<span class="_ _6"> </span><span class="ff3">A<span class="_ _7"> </span></span>=</div><div class="t m0 xf h6 y54 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x74 h5 y32b ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 xb5 h5 y242 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x6a h6 y54 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x6a h5 y243 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=1</span></div><div class="t m0 x13 h3 y24a ff2 fs0 fc0 sc0 ls0 ws0">sign(<span class="ff3">π<span class="_ _3"></span></span>)<span class="ff3">a</span></div><div class="t m0 x122 h5 y32c ff6 fs1 fc0 sc0 ls0 ws0">i,π<span class="_ _3"></span><span class="ff5">(</span>i<span class="ff5">)</span></div><div class="t m0 x17 h3 y24a ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x153 h4 y32d ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _8"> </span><span class="ff4">×<span class="_ _8"> </span></span>2 and 3<span class="_ _8"> </span><span class="ff4">×<span class="_ _8"> </span></span>3 determinan<span class="_ _5"></span>t:</div><div class="t m0 xb4 h6 y32e ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb4 h6 y24 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb4 h6 y24c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb4 h6 y32f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x3b h3 y330 ff3 fs0 fc0 sc0 ls16 ws0">ab</div><div class="t m0 x3b h3 y331 ff3 fs0 fc0 sc0 ls17 ws0">cd</div><div class="t m0 x11 h6 y332 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x11 h6 y24 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x11 h6 y24c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x11 h6 y333 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb5 h4 y25 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">ad<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>bc,</span></div><div class="t m0 xb h6 y334 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb h6 y335 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb h6 y336 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb h6 y157 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb h6 y337 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb h6 y158 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x15e h3 y338 ff3 fs0 fc0 sc0 ls18 ws0">ab<span class="_ _15"></span>c</div><div class="t m0 x15e h3 y142 ff3 fs0 fc0 sc0 ls18 ws0">def</div><div class="t m0 x15e h3 y339 ff3 fs0 fc0 sc0 ls16 ws0">gh<span class="_ _2"></span>i</div><div class="t m0 x5b h6 y33a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x5b h6 y33b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x5b h6 y33c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x5b h6 y157 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x5b h6 y33d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x5b h6 y158 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xff h3 y142 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">g</span></div><div class="t m0 x196 h6 y33e ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x196 h6 y33f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x196 h6 y157 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x196 h6 y337 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x170 h3 y5e ff3 fs0 fc0 sc0 ls19 ws0">bc</div><div class="t m0 x170 h3 y340 ff3 fs0 fc0 sc0 ls5 ws0">ef</div><div class="t m0 xaf h6 y33b ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xaf h6 y336 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xaf h6 y157 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xaf h6 y337 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x2e h4 y142 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">h</span></div><div class="t m0 xcd h6 y335 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xcd h6 y336 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xcd h6 y157 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xcd h6 y337 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x3f h3 y5e ff3 fs0 fc0 sc0 ls19 ws0">ac</div><div class="t m0 x3f h3 y340 ff3 fs0 fc0 sc0 ls5 ws0">df</div><div class="t m0 x9f h6 y335 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x9f h6 y336 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x9f h6 y157 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x9f h6 y337 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x30 h3 y142 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">i</span></div><div class="t m0 x186 h6 y335 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x186 h6 y336 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x186 h6 y157 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x186 h6 y337 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x112 h3 y5e ff3 fs0 fc0 sc0 ls5 ws0">ab</div><div class="t m0 x112 h3 y340 ff3 fs0 fc0 sc0 ls5 ws0">de</div><div class="t m0 x60 h6 y335 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x60 h6 y336 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x60 h6 y157 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x60 h6 y337 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xff h3 y341 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xa8 h3 y342 ff3 fs0 fc0 sc0 ls0 ws0">aei<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>bf<span class="_ _2"></span>g<span class="_ _7"> </span><span class="ff2">+<span class="_ _8"> </span></span>cdh</div><div class="t m0 x75 h4 y343 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">ceg<span class="_ _7"> </span></span>−<span class="_ _8"></span><span class="ff3 ls1a">fh<span class="_ _1f"></span>a<span class="_ _2"></span><span class="ff4 ls0">−<span class="_ _8"> </span><span class="ff3">ibd.</span></span></span></div><div class="t m0 x153 h3 y344 ff2 fs0 fc0 sc0 ls0 ws0">P<span class="_ _5"></span>ermanents:</div><div class="t m0 x2a h3 y345 ff2 fs0 fc0 sc0 ls0 ws0">p<span class="_ _3"></span>erm<span class="_ _6"> </span><span class="ff3">A<span class="_ _7"> </span></span>=</div><div class="t m0 x197 h6 y346 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xae h5 y347 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x198 h5 y348 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x5d h6 y349 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x5d h5 y347 ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=1</span></div><div class="t m0 x3f h3 y34a ff3 fs0 fc0 sc0 ls0 ws0">a</div><div class="t m0 x6c h5 y15b ff6 fs1 fc0 sc0 ls0 ws0">i,π<span class="_ _3"></span><span class="ff5">(</span>i<span class="ff5">)</span></div><div class="t m0 x31 h3 y34a ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x1d h3 y34b ff3 fs0 fc0 sc0 ls0 ws0">A</div><div class="t m0 xe0 h3 y34c ff3 fs0 fc0 sc0 ls0 ws0">a</div><div class="t m0 xbd h3 y34b ff3 fs0 fc0 sc0 ls0 ws0">c</div><div class="t m0 x20 h3 y122 ff3 fs0 fc0 sc0 ls0 ws0">h</div><div class="t m0 xcf h3 y23b ff3 fs0 fc0 sc0 ls0 ws0">b</div><div class="t m0 x38 h3 y34b ff3 fs0 fc0 sc0 ls0 ws0">B</div><div class="t m0 x37 h3 y34d ff3 fs0 fc0 sc0 ls0 ws0">C</div><div class="t m0 x51 h3 y53 ff2 fs0 fc0 sc0 ls0 ws0">La<span class="_ _5"></span>w of cosines:</div><div class="t m0 x51 h3 y34e ff3 fs0 fc0 sc0 ls0 ws0">c</div><div class="t m0 x70 h5 y34f ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xb1 h3 y350 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">a</span></div><div class="t m0 x53 h5 y34f ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1f h3 y350 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="ff3">b</span></div><div class="t m0 x65 h5 y34f ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa4 h4 y350 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="ff2">2<span class="ff3">ab<span class="_ _6"> </span></span>cos<span class="_ _8"> </span><span class="ff3 ls10">C.</span></span></div><div class="t m0 x51 h3 y134 ff2 fs0 fc0 sc0 ls0 ws0">Area:</div><div class="t m0 x70 h3 y351 ff3 fs0 fc0 sc0 ls0 ws0">A<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x53 h5 y32e ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x53 h5 y352 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x199 h3 y353 ff3 fs0 fc0 sc0 ls0 ws0">hc,</div><div class="t m0 x64 h3 y354 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x53 h5 y355 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x53 h5 y29 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x199 h3 y356 ff3 fs0 fc0 sc0 ls0 ws0">ab<span class="_ _6"> </span><span class="ff2">sin<span class="_ _8"> </span></span><span class="ls10">C,</span></div><div class="t m0 x64 h3 y13f ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x53 h3 y141 ff3 fs0 fc0 sc0 ls0 ws0">c</div><div class="t m0 x37 h5 y357 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xf0 h3 y141 ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">A<span class="_ _8"> </span></span>sin<span class="_ _2"> </span><span class="ff3">B</span></div><div class="t m0 xb2 h3 y142 ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _6"> </span>sin<span class="_ _8"> </span><span class="ff3">C</span></div><div class="t m0 x7f h3 y13f ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x51 h3 y5f ff2 fs0 fc0 sc0 ls0 ws0">Heron’s form<span class="_ _5"></span>ula:</div><div class="t m0 x63 h3 y39 ff3 fs0 fc0 sc0 ls0 ws0">A<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 xbc h4 y358 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 x1f h4 y39 ff3 fs0 fc0 sc0 ls0 ws0">s<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>s</div><div class="t m0 x66 h5 y359 ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 x7d h4 y39 ff4 fs0 fc0 sc0 ls0 ws0">·<span class="_ _8"> </span><span class="ff3">s</span></div><div class="t m0 xba h5 y359 ff6 fs1 fc0 sc0 ls0 ws0">b</div><div class="t m0 x184 h4 y39 ff4 fs0 fc0 sc0 ls0 ws0">·<span class="_ _8"> </span><span class="ff3">s</span></div><div class="t m0 x185 h5 y359 ff6 fs1 fc0 sc0 ls0 ws0">c</div><div class="t m0 xa7 h3 y39 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x52 h3 y2fc ff3 fs0 fc0 sc0 ls0 ws0">s<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 xd0 h5 y3e ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xd0 h5 y35a ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x37 h3 y2fc ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff3">a<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">b<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">c</span>)<span class="ff3">,</span></div><div class="t m0 x17e h3 y285 ff3 fs0 fc0 sc0 ls0 ws0">s</div><div class="t m0 x52 h5 y35b ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 x1d h4 y285 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">s<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>a,</span></div><div class="t m0 x63 h3 y35c ff3 fs0 fc0 sc0 ls0 ws0">s</div><div class="t m0 x52 h5 y35d ff6 fs1 fc0 sc0 ls0 ws0">b</div><div class="t m0 x1d h4 y35e ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">s<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>b,</span></div><div class="t m0 x63 h3 y35f ff3 fs0 fc0 sc0 ls0 ws0">s</div><div class="t m0 x52 h5 y360 ff6 fs1 fc0 sc0 ls0 ws0">c</div><div class="t m0 x1d h4 y361 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">s<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>c.</span></div><div class="t m0 x51 h3 y362 ff2 fs0 fc0 sc0 ls0 ws0">More iden<span class="_ _5"></span>tities:</div><div class="t m0 x63 h3 y363 ff2 fs0 fc0 sc0 ls0 ws0">sin</div><div class="t m0 xcf h5 y1ae ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 xcf h5 y28b ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x53 h3 y364 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x20 h6 y300 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x66 h4 y365 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 xba h3 y366 ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x19a h3 y367 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x63 h3 y7d ff2 fs0 fc0 sc0 ls0 ws0">cos</div><div class="t m0 xcf h5 y305 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 xcf h5 y306 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x53 h3 y7d ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x20 h6 y368 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x66 h3 y7a ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span>+<span class="_ _8"> </span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 xba h3 y369 ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x19a h3 y7d ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x17e h3 y36a ff2 fs0 fc0 sc0 ls0 ws0">tan</div><div class="t m0 xcf h5 y36b ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 xcf h5 y36c ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x53 h3 y36d ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x20 h6 y87 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x66 h4 y36e ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x66 h3 y36f ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span>+<span class="_ _8"> </span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x19a h3 y370 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x53 h3 y371 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x20 h4 y372 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 xa4 h3 y291 ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x19b h3 y373 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x53 h3 y99 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xa4 h3 y374 ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x20 h3 y375 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span>+<span class="_ _8"> </span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x19b h3 y99 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x63 h3 y294 ff2 fs0 fc0 sc0 ls0 ws0">cot</div><div class="t m0 xcf h5 yb6 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 xcf h5 y181 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x53 h3 y294 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x20 h6 y293 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x66 h3 y206 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span>+<span class="_ _8"> </span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x66 h4 yb7 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x19a h3 y294 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x53 h3 y376 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x20 h3 y377 ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span>+<span class="_ _8"> </span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 xa4 h3 y378 ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x19b h3 y379 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x53 h3 y186 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xa4 h3 y37a ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x20 h4 y18a ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>cos<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x19b h3 y186 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 xde h3 y29b ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=</div><div class="t m0 x20 h3 y37b ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x65 h5 y2db ff6 fs1 fc0 sc0 ls0 ws0">ix</div><div class="t m0 xbe h4 y37c ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">e</span></div><div class="t m0 x105 h5 y2db ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">ix</span></div><div class="t m0 x7d h3 y1dc ff2 fs0 fc0 sc0 ls0 ws0">2<span class="ff3">i</span></div><div class="t m0 x19c h3 y29b ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x70 h3 y2a3 ff2 fs0 fc0 sc0 ls0 ws0">cos<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=</div><div class="t m0 x20 h3 y1da ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x65 h5 y18f ff6 fs1 fc0 sc0 ls0 ws0">ix</div><div class="t m0 xbe h3 y1da ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">e</span></div><div class="t m0 x105 h5 y18f ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">ix</span></div><div class="t m0 xe0 h3 y193 ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x19c h3 y2a3 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x63 h4 y217 ff2 fs0 fc0 sc0 ls0 ws0">tan<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=<span class="_ _7"> </span><span class="ff4">−<span class="ff3">i</span></span></div><div class="t m0 x66 h3 y37d ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 xa6 h5 y37e ff6 fs1 fc0 sc0 ls0 ws0">ix</div><div class="t m0 xa5 h4 y37f ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">e</span></div><div class="t m0 x24 h5 y37e ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">ix</span></div><div class="t m0 x66 h3 y199 ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 xa6 h5 y380 ff6 fs1 fc0 sc0 ls0 ws0">ix</div><div class="t m0 xa5 h3 y199 ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">e</span></div><div class="t m0 x24 h5 y380 ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">ix</span></div><div class="t m0 x19d h3 y217 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x53 h4 y381 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−<span class="ff3">i</span></span></div><div class="t m0 x66 h3 y382 ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 xa6 h5 y383 ff5 fs1 fc0 sc0 ls0 ws0">2<span class="ff6">ix</span></div><div class="t m0 x165 h4 y384 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">1</span></div><div class="t m0 x66 h3 ye9 ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 xa6 h5 y19b ff5 fs1 fc0 sc0 ls0 ws0">2<span class="ff6">ix</span></div><div class="t m0 x165 h3 ye9 ff2 fs0 fc0 sc0 ls2 ws0">+1</div><div class="t m0 x39 h3 y385 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 xde h3 y386 ff2 fs0 fc0 sc0 ls0 ws0">sin<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=</div><div class="t m0 x20 h3 y387 ff2 fs0 fc0 sc0 ls0 ws0">sinh<span class="_ _6"> </span><span class="ff3">ix</span></div><div class="t m0 xd1 h3 y388 ff3 fs0 fc0 sc0 ls0 ws0">i</div><div class="t m0 x165 h3 y389 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x70 h3 yfa ff2 fs0 fc0 sc0 ls0 ws0">cos<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=<span class="_ _7"> </span>cosh<span class="_ _8"> </span><span class="ff3">ix,</span></div><div class="t m0 x63 h3 y19f ff2 fs0 fc0 sc0 ls0 ws0">tan<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=</div><div class="t m0 x20 h3 y38a ff2 fs0 fc0 sc0 ls0 ws0">tanh<span class="_ _6"> </span><span class="ff3">ix</span></div><div class="t m0 x23 h3 y38b ff3 fs0 fc0 sc0 ls0 ws0">i</div><div class="t m0 xc0 h3 y19f ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 xe h3 y360 ff2 fs0 fc0 sc0 ls0 ws0">Hyp<span class="_ _3"></span>erb<span class="_ _3"></span>olic F<span class="_ _5"></span>unctions</div><div class="t m0 x19e h3 y6d ff2 fs0 fc0 sc0 ls0 ws0">Definitions:</div><div class="t m0 x25 h3 y1af ff2 fs0 fc0 sc0 ls0 ws0">sinh<span class="_ _6"> </span><span class="ff3">x<span class="_ _0"> </span></span>=</div><div class="t m0 x19f h3 y38c ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x2b h5 y6e ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x5b h4 y38d ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">e</span></div><div class="t m0 x9e h5 y6e ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">x</span></div><div class="t m0 xe h3 y38e ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 xf h3 y1af ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _54"> </span><span class="ff2">cosh<span class="_ _6"> </span></span>x<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x41 h3 y38f ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x17f h5 y6e ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x31 h3 y38d ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">e</span></div><div class="t m0 x162 h5 y6e ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">x</span></div><div class="t m0 x30 h3 y38e ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x6d h3 y1af ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x19e h3 y7a ff2 fs0 fc0 sc0 ls0 ws0">tanh<span class="_ _6"> </span><span class="ff3">x<span class="_ _7"> </span></span>=</div><div class="t m0 x19f h3 y7c ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x2b h5 y390 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x5b h4 y7c ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">e</span></div><div class="t m0 x9e h5 y390 ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">x</span></div><div class="t m0 x19f h3 y7d ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x2b h5 y1f9 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x5b h3 y7d ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff3">e</span></div><div class="t m0 x9e h5 y1f9 ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">x</span></div><div class="t m0 xf h3 y7a ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _54"> </span><span class="ff2">csc<span class="_ _5"></span>h<span class="_ _8"> </span><span class="ff3">x </span>=</span></div><div class="t m0 x9f h3 y7c ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x41 h3 y7d ff2 fs0 fc0 sc0 ls0 ws0">sinh<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x162 h3 y7a ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x25 h3 y391 ff2 fs0 fc0 sc0 ls0 ws0">sec<span class="_ _5"></span>h<span class="_ _8"> </span><span class="ff3">x </span>=</div><div class="t m0 x14e h3 y83 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x19f h3 y36e ff2 fs0 fc0 sc0 ls0 ws0">cosh<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 xb4 h3 y392 ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _2a"> </span><span class="ff2">coth<span class="_ _6"> </span></span>x<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x31 h3 y83 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x41 h3 y36e ff2 fs0 fc0 sc0 ls0 ws0">tanh<span class="_ _6"> </span><span class="ff3">x</span></div><div class="t m0 x33 h3 y392 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x19e h3 y393 ff2 fs0 fc0 sc0 ls0 ws0">Iden<span class="_ _5"></span>tities:</div><div class="t m0 x19e h3 y394 ff2 fs0 fc0 sc0 ls0 ws0">cosh</div><div class="t m0 x12e h5 y1fe ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xc h4 y395 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">sinh</span></span></div><div class="t m0 xff h5 y1fe ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x178 h3 y395 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _7"> </span><span class="ff2 ls1">=1<span class="_ _b"></span><span class="ff3 ls0">,<span class="_ _55"> </span><span class="ff2">tanh</span></span></span></div><div class="t m0 x6c h5 y1fe ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x41 h3 y395 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span>sec<span class="_ _5"></span>h</span></div><div class="t m0 x4d h5 y1fe ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x11d h3 y395 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _7"> </span><span class="ff2 ls1">=1<span class="_ _b"></span><span class="ff3 ls0">,</span></span></div><div class="t m0 x19e h3 y1c4 ff2 fs0 fc0 sc0 ls0 ws0">coth</div><div class="t m0 x12e h5 y172 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xc h4 y1c4 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">csc<span class="_ _5"></span>h</span></span></div><div class="t m0 x126 h5 y172 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x9e h4 y1c4 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _7"> </span><span class="ff2 ls1">=1<span class="_ _b"></span><span class="ff3 ls0">,<span class="_ _4d"> </span><span class="ff2">sinh(<span class="_ _5"></span><span class="ff4">−<span class="ff3">x<span class="ff2 ls1">)=</span></span>−<span class="_ _6"></span><span class="ff2">sinh<span class="_ _8"> </span><span class="ff3">x,</span></span></span></span></span></span></div><div class="t m0 x19e h4 y9f ff2 fs0 fc0 sc0 ls0 ws0">cosh(<span class="ff4">−<span class="ff3">x</span></span>)<span class="_ _7"> </span>=<span class="_ _7"> </span>cosh<span class="_ _6"> </span><span class="ff3">x,<span class="_ _4a"> </span></span>tanh(<span class="ff4">−<span class="ff3">x</span></span><span class="ls1">)=</span><span class="ff4">−<span class="_ _6"></span></span>tanh<span class="_ _8"> </span><span class="ff3">x,</span></div><div class="t m0 x19e h3 y396 ff2 fs0 fc0 sc0 ls0 ws0">sinh(<span class="ff3">x<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">y<span class="_ _3"></span></span>)<span class="_ _7"> </span>=<span class="_ _7"> </span>sinh<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>cosh<span class="_ _2"> </span><span class="ff3">y<span class="_ _7"> </span></span>+<span class="_ _8"> </span>cosh<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>sinh<span class="_ _2"> </span><span class="ff3 ls10">y,</span></div><div class="t m0 x19e h3 y2d5 ff2 fs0 fc0 sc0 ls0 ws0">cosh(<span class="ff3">x<span class="_ _8"> </span></span>+<span class="_ _8"> </span><span class="ff3">y<span class="_ _3"></span></span>)<span class="_ _7"> </span>=<span class="_ _7"> </span>cosh<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>cosh<span class="_ _6"> </span><span class="ff3">y<span class="_ _8"> </span></span>+<span class="_ _8"> </span>sinh<span class="_ _8"> </span><span class="ff3">x<span class="_ _2"> </span></span>sinh<span class="_ _8"> </span><span class="ff3 ls10">y,</span></div><div class="t m0 x19e h3 y2ea ff2 fs0 fc0 sc0 ls0 ws0">sinh<span class="_ _6"> </span>2<span class="ff3">x<span class="_ _7"> </span></span>=<span class="_ _7"> </span>2<span class="_ _8"> </span>sinh<span class="_ _2"> </span><span class="ff3">x<span class="_ _8"> </span></span>cosh<span class="_ _2"> </span><span class="ff3">x,</span></div><div class="t m0 x19e h3 y397 ff2 fs0 fc0 sc0 ls0 ws0">cosh<span class="_ _6"> </span>2<span class="ff3">x<span class="_ _7"> </span></span>=<span class="_ _7"> </span>cosh</div><div class="t m0 x68 h5 y213 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x15f h3 y398 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span>sinh</span></div><div class="t m0 xae h5 y213 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x13 h3 y398 ff3 fs0 fc0 sc0 ls0 ws0">x,</div><div class="t m0 x19e h3 y399 ff2 fs0 fc0 sc0 ls0 ws0">cosh<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>+<span class="_ _8"> </span>sinh<span class="_ _8"> </span><span class="ff3">x<span class="_ _8"> </span></span>=<span class="_ _7"> </span><span class="ff3">e</span></div><div class="t m0 xf h5 y191 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x10e h4 y39a ff3 fs0 fc0 sc0 ls0 ws0">,<span class="_ _18"> </span><span class="ff2">cosh<span class="_ _8"> </span></span>x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"></span><span class="ff2">sinh<span class="_ _2"> </span></span></span>x<span class="_ _7"> </span><span class="ff2">=<span class="_ _7"> </span></span>e</div><div class="t m0 xa1 h5 y191 ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">x</span></div><div class="t m0 x50 h3 y39a ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x19e h3 y39b ff2 fs0 fc0 sc0 ls0 ws0">(cosh<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span>+<span class="_ _8"> </span>sinh<span class="_ _8"> </span><span class="ff3">x<span class="_ _5"></span><span class="ff2">)</span></span></div><div class="t m0 x5c h5 y39c ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x5f h4 y39d ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>cosh<span class="_ _6"> </span><span class="ff3">nx<span class="_ _8"> </span></span>+<span class="_ _8"> </span>sinh<span class="_ _8"> </span><span class="ff3">nx,<span class="_ _c"> </span>n<span class="_ _8"> </span><span class="ff4">∈<span class="_ _7"> </span><span class="ff9">Z</span></span>,</span></div><div class="t m0 x19e h3 y31d ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _6"> </span>sinh</div><div class="t m0 xc h5 y39e ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x46 h5 y2ed ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x46 h5 y1e2 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xb3 h4 y31d ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>cosh<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span></span>1<span class="ff3">,<span class="_ _56"> </span></span>2<span class="_ _6"> </span>cosh</div><div class="t m0 x40 h5 y39e ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x77 h5 y2ed ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x77 h5 y1e2 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x9f h3 y31d ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span>cosh<span class="_ _6"> </span><span class="ff3">x<span class="_ _8"> </span></span><span class="ls2">+1<span class="_ _9"></span><span class="ff3 ls0">.</span></span></div><div class="t m0 x44 h3 y19b ff3 fs0 fc0 sc0 ls0 ws0">θ<span class="_ _c"> </span><span class="ff2">sin<span class="_ _6"> </span></span>θ<span class="_ _28"> </span><span class="ff2">cos<span class="_ _8"> </span></span>θ<span class="_ _28"> </span><span class="ff2">tan<span class="_ _6"> </span></span>θ</div><div class="t m0 x44 h3 y39f ff2 fs0 fc0 sc0 ls1b ws0">00<span class="_ _21"> </span>1<span class="_ _e"> </span>0</div><div class="t m0 x44 h5 y2ba ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x44 h5 yfc ff5 fs1 fc0 sc0 ls0 ws0">6</div><div class="t m0 x72 h5 y2ba ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x72 h5 yfc ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2a h5 y2b5 ff8 fs1 fc0 sc0 ls0 ws0">√</div><div class="t m0 x126 h5 y2ba ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 xe h5 yfc ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x74 h5 y2b5 ff8 fs1 fc0 sc0 ls0 ws0">√</div><div class="t m0 x75 h5 y2ba ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 xa8 h5 yfc ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x44 h5 y2e6 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x44 h5 y3a0 ff5 fs1 fc0 sc0 ls0 ws0">4</div><div class="t m0 x59 h5 y2bc ff8 fs1 fc0 sc0 ls0 ws0">√</div><div class="t m0 x137 h5 y2e6 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x72 h5 y3a0 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2a h5 y2bc ff8 fs1 fc0 sc0 ls0 ws0">√</div><div class="t m0 x126 h5 y2e6 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xe h5 y3a0 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xa8 h3 y3a1 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x44 h5 y3a2 ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x44 h5 y38b ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x59 h5 y38a ff8 fs1 fc0 sc0 ls0 ws0">√</div><div class="t m0 x137 h5 y3a2 ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x72 h5 y38b ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xe h5 y3a3 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xe h5 y38b ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x15c h4 y3a4 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 x75 h3 y10b ff2 fs0 fc0 sc0 ls0 ws0">3</div><div class="t m0 x44 h5 y2be ff6 fs1 fc0 sc0 ls0 ws0">π</div><div class="t m0 x44 h5 y1ec ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x16f h4 y3a5 ff2 fs0 fc0 sc0 ls1c ws0">10<span class="_ _1f"></span><span class="ff4 ls0">∞</span></div><div class="t m0 x6b h3 y19b ff3 fs0 fc0 sc0 ls4 ws0">...<span class="_ _2"></span><span class="ff2 ls0">in<span class="_ _7"> </span>mathematics</span></div><div class="t m0 x6b h3 y2b3 ff2 fs0 fc0 sc0 ls0 ws0">y<span class="_ _5"></span>ou<span class="_ _e"> </span>don’t<span class="_ _36"> </span>under-</div><div class="t m0 x6b h3 y3a6 ff2 fs0 fc0 sc0 ls0 ws0">stand<span class="_ _11"> </span>things,<span class="_ _21"> </span>you</div><div class="t m0 x6b h3 y3a7 ff2 fs0 fc0 sc0 ls0 ws0">just<span class="_ _e"> </span>get<span class="_ _e"> </span>used<span class="_ _e"> </span>to</div><div class="t m0 x6b h3 y3a8 ff2 fs0 fc0 sc0 ls0 ws0">them.</div><div class="t m0 x6b h3 y3a9 ff2 fs0 fc0 sc0 ls0 ws0">– J. v<span class="_ _5"></span>on Neumann</div><div class="t m0 x18c h3 y3a9 ff2 fs0 fc0 sc0 ls0 ws0">v2.02</div><div class="t m0 x131 h3 y3a9 ff2 fs0 fc0 sc0 ls0 ws0">c</div><div class="t m0 xf2 h4 y3a9 ff4 fs0 fc0 sc0 ls0 ws0"><span class="ff2">1994 b<span class="_ _5"></span>y Steve Seiden</span></div><div class="t m0 x120 h2 y3aa ffe fs0 fc0 sc0 ls0 ws0">sseiden@acm.org</div><div class="t m0 xe1 h2 y10a ffe fs0 fc0 sc0 ls0 ws0">http://www.csc.lsu.edu/~<span class="_ _5"></span>seiden</div></div><div class="pi" data-data='{"ctm":[1.673203,0.000000,0.000000,1.673203,0.000000,0.000000]}'></div></div></div>