Trigonometry Cheat Sheet


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Basic Identities

\tan(x) = \frac{\sin(x)}{\cos(x)} \tan(x) = \frac{1}{\cot(x)}
\cot(x) = \frac{1}{\tan(x)} \cot(x) = \frac{\cos(x)}{\sin(x)}
\sec(x) = \frac{1}{\cos(x)} \csc(x) = \frac{1}{\sin(x)}


Pythagorean Identities

\cos^2(x)+\sin^2(x) = 1 \sec^2(x)-\tan^2(x) = 1
\csc^2(x)-\cot^2(x) = 1


Double-Angle Identities

\sin(2x)=2\sin(x)\cos(x) \cos(2x)=1-2\sin^2(x)
\cos(2x) = 2\cos^2(x)-1 \cos(2x) = \cos^2(x)-\sin^2(x)
\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}


Sum/Difference Identities

\sin(s+t) = \sin(s)\cos(t)+\cos(s)\sin(t)
\sin(s-t) = \sin(s)\cos(t)-\cos(s)\sin(t)
\cos(s+t) = \cos(s)\cos(t)-\sin(s)\sin(t)
\cos(s-t) = \cos(s)\cos(t)+\sin(s)\sin(t)
\tan(s+t) = \frac{\tan(s)+\tan(t)}{1-\tan(s)\tan(t)}
\tan(s-t) = \frac{\tan(s)-\tan(t)}{1+\tan(s)\tan(t)}


Product-To-Sum Identities

\cos(s)\cos(t)=\frac{\cos(s-t)+\cos(s+t)}{2}
\sin(s)\sin(t)=\frac{\cos(s-t)-\cos(s+t)}{2}
\sin(s)\cos(t)=\frac{\sin(s+t)+\sin(s-t)}{2}
\cos(s)\sin(t)=\frac{\sin(s+t)-\sin(s-t)}{2}


Triple-Angle Identities

\sin(3x)=-\sin^3(x)+3\cos^2(x)\sin(x)
\sin(3x)=-4\sin^3(x)+3\sin(x)
\cos(3x)=\cos^3(x)-3\sin^2(x)\cos(x)
\cos(3x)=4\cos^3(x)-3\cos(x)
\tan(3x)=\frac{3\tan(x)-\tan^3(x)}{1-3\tan^2(x)}
\cot(3x)=\frac{3\cot(x)-\cot^3(x)}{1-3\cot^2(x)}


Function Ranges

y = \sin(x) -1\le y\le 1
y = \cos(x) -1\le y\le 1
y = \tan(x) -\infty < y <\infty
y = \cot(x) -\infty < y <\infty
y = \csc(x) -\infty < y\le 1\:\bigcup \:1\le y < \infty
y = \sec(y) -\infty < y\le 1\:\bigcup \:1\le y < \infty
y = \arcsin(x) -\frac{\pi \:}{2}\:\le y\le \:\:\frac{\pi \:}{2}\:
y = \arccos(x) 0\:\le \:y\:\le \:\pi
y = \arctan(x) -\frac{\pi \:}{2} < \:y < \frac{\pi \:}{2}:
y = \arccot(x) 0 < x < \pi
y = \arccsc(x) 0\le y <\frac{\pi }{2}\:\bigcup \:\pi\le y <\frac{3\pi }{2}
y = \arcsec(x) -\pi < y\le -\frac{\pi }{2}\:\bigcup \:0 < y < \frac{\pi }{2}<\infty


Function Values

sin(x) cos(x) tan(x) cot(x)
0 0 1 0 \text{Undefined}
\frac{π}{6} \frac{1}{2} \frac{\sqrt{3}}{2} \frac{\sqrt{3}}{3} \sqrt{3}
\frac{π}{4} \frac{\sqrt{2}}{2} \frac{\sqrt{2}}{2} 1 1
\frac{π}{3} \frac{\sqrt{3}}{2} \frac{1}{2} \sqrt{3} \frac{\sqrt{3}}{3}
\frac{π}{2} 1 0 \mathrm{Undefined} 0
\frac{2π}{3} \frac{\sqrt{3}}{2} -\frac{1}{2} -\sqrt{3} -\frac{\sqrt{3}}{3}
\frac{3π}{4} \frac{\sqrt{2}}{2} -\frac{\sqrt{2}}{2} -1 -1
\frac{5π}{6} \frac{1}{2} -\frac{\sqrt{3}}{2} -\frac{\sqrt{3}}{3} -\sqrt{3}
π 0 -1 0 \mathrm{Undefined}
\frac{7π}{6} -\frac{1}{2} -\frac{\sqrt{3}}{2}
\frac{5π}{4} -\frac{\sqrt{2}}{2} -\frac{\sqrt{2}}{2}
\frac{4π}{3} -\frac{\sqrt{3}}{2} -\frac{1}{2}
\frac{3π}{2} -1 0
\frac{5π}{3} -\frac{\sqrt{3}}{2} \frac{1}{2}
\frac{7π}{4} -\frac{\sqrt{2}}{2} \frac{\sqrt{2}}{2}
\frac{11π}{6} -\frac{1}{2} \frac{\sqrt{3}}{2}
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