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1070 </head>
1072 <body class="markdown-preview-plus-view">
1073 <h1>Circular functions</h1>
1074<h2>Radians and degrees</h2>
1075<span class="math"><script type="math/tex; mode=display">1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}
1076</script></span>
1077<h2>Exact values</h2>
1078<h2><span class="math"><script type="math/tex">\sin</script></span> and <span class="math"><script type="math/tex">\cos</script></span> graphs</h2>
1079<span class="math"><script type="math/tex; mode=display">f(x)=a \sin(bx-c)+d
1080</script></span>
1081<span class="math"><script type="math/tex; mode=display">f(x)=a \cos(bx-c)+d
1082</script></span>
1083<p>where<br>
1084<span class="math"><script type="math/tex">a</script></span> is the <span class="math"><script type="math/tex">y</script></span>-dilation (amplitude)<br>
1085<span class="math"><script type="math/tex">b</script></span> is the <span class="math"><script type="math/tex">x</script></span>-dilation (period)<br>
1086<span class="math"><script type="math/tex">c</script></span> is the <span class="math"><script type="math/tex">x</script></span>-shift (phase)<br>
1087<span class="math"><script type="math/tex">d</script></span> is the <span class="math"><script type="math/tex">y</script></span>-shift (equilibrium position)</p>
1088<p>Domain is <span class="math"><script type="math/tex">\mathbb{R}</script></span><br>
1089Range is <span class="math"><script type="math/tex">[-b+c, b+c]</script></span>;</p>
1090<p>Graph of <span class="math"><script type="math/tex">\cos(x)</script></span> starts at <span class="math"><script type="math/tex">(0,1)</script></span>. Graph of <span class="math"><script type="math/tex">\sin(x)</script></span> starts at <span class="math"><script type="math/tex">(0,0)</script></span>.</p>
1091<p><strong>Mean / equilibrium:</strong> line that the graph oscillates around (<span class="math"><script type="math/tex">y=d</script></span>)</p>
1092<h2>Solving trig equations</h2>
1093<ol>
1094<li>Solve domain for <span class="math"><script type="math/tex">n\theta</script></span></li>
1095<li>Find solutions for <span class="math"><script type="math/tex">n\theta</script></span></li>
1096<li>Divide solutions by <span class="math"><script type="math/tex">n</script></span></li>
1097</ol>
1098<p><span class="math"><script type="math/tex">\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])</script></span><br>
1099<span class="math"><script type="math/tex">2\theta=\sin^{-1}{\sqrt{3} \over 2}</script></span><br>
1100<span class="math"><script type="math/tex">2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}</script></span><br>
1101<span class="math"><script type="math/tex">\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}</script></span></p>
1102<h3>Amplitude</h3>
1103<p>Amplitude of <span class="math"><script type="math/tex">a</script></span> means graph oscillates between <span class="math"><script type="math/tex">+a</script></span> and <span class="math"><script type="math/tex">-a</script></span> in <span class="math"><script type="math/tex">y</script></span>-axis</p>
1104<p><span class="math"><script type="math/tex">a=0</script></span> produces straight line<br>
1105<span class="math"><script type="math/tex">a\lt0</script></span> inverts the phase (<span class="math"><script type="math/tex">\sin</script></span> becomes <span class="math"><script type="math/tex">\cos</script></span>, vice vera)</p>
1106<h3>Period</h3>
1107<p>Period <span class="math"><script type="math/tex">T</script></span> is <span class="math"><script type="math/tex">{2 \pi}\over b</script></span><br>
1108<span class="math"><script type="math/tex">b=0</script></span> produces straight line<br>
1109<span class="math"><script type="math/tex">b\lt0</script></span> inverts the phase</p>
1110<h3>Phase</h3>
1111<p><span class="math"><script type="math/tex">c</script></span> moves the graph left-right in the <span class="math"><script type="math/tex">x</script></span> axis.<br>
1112If <span class="math"><script type="math/tex">c=T={{2\pi}\over b}</script></span>, the graph has no actual phase shift.</p>
1113<h2>Symmetry</h2>
1114<span class="math"><script type="math/tex; mode=display">\sin(\theta+{\pi\over 2})=\sin\theta
1115</script></span>
1116<span class="math"><script type="math/tex; mode=display">\sin(\theta+\pi)=-\sin\theta
1117</script></span>
1118<span class="math"><script type="math/tex; mode=display">\cos(\theta+{\pi \over 2})=-\cos\theta
1119</script></span>
1120<span class="math"><script type="math/tex; mode=display">\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)
1121</script></span>
1122<h2>Pythagorean identity</h2>
1123<span class="math"><script type="math/tex; mode=display">\cos^2\theta+\sin^2\theta=1
1124</script></span>
1125<h2>Complementary relationships</h2>
1126<span class="math"><script type="math/tex; mode=display">\sin({\pi \over 2} - \theta)=\cos\theta
1127</script></span>
1128<span class="math"><script type="math/tex; mode=display">\cos({\pi \over 2} - \theta)=\sin\theta
1129</script></span>
1130<span class="math"><script type="math/tex; mode=display">\sin\theta=-\cos(\theta+{\pi \over 2})
1131</script></span>
1132<span class="math"><script type="math/tex; mode=display">\cos\theta=\sin(\theta+{\pi \over 2})
1133</script></span>
1134<h2><span class="math"><script type="math/tex">\tan</script></span> graph</h2>
1135<span class="math"><script type="math/tex; mode=display">y=a\tan(nx)
1136</script></span>
1137<p>where<br>
1138<span class="math"><script type="math/tex">a</script></span> is <span class="math"><script type="math/tex">x</script></span>-dilation (period)<br>
1139<span class="math"><script type="math/tex">n</script></span> is <span class="math"><script type="math/tex">y</script></span>-dilation (<span class="math"><script type="math/tex">\equiv</script></span> amplitude)<br>
1140period <span class="math"><script type="math/tex">T</script></span> is <span class="math"><script type="math/tex">\pi \over n</script></span><br>
1141range is <span class="math"><script type="math/tex">R</script></span><br>
1142roots at <span class="math"><script type="math/tex">x={k\pi \over n}</script></span><br>
1143asymptotes at <span class="math"><script type="math/tex">x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}</script></span><br>
1144<strong>Asymptotes should always have equations and arrow pointing up</strong></p>
1145 </body>
1147</html>