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Section B.3 Trigonometry Reference
Subsection B.3.1 Definitions of the Trigonometric Functions
Unit Circle Definition.
| \(\sin(\theta) = y\) |
\(\cos(\theta) = x\) |
|
| \(\ds\csc(\theta) = \frac1y\) |
\(\ds\sec(\theta) = \frac1x\) |
|
| \(\ds\tan(\theta) = \frac yx\) |
\(\ds\cot(\theta) = \frac xy\) |
Right Triangle Definition.
| \(\ds\sin(\theta) = \frac{\text{O} }{\text{H} }\) |
\(\ds\csc(\theta) = \frac{\text{H} }{\text{O} }\) |
|
| \(\ds\cos(\theta) = \frac{\text{A} }{\text{H} }\) |
\(\ds\sec(\theta) = \frac{\text{H} }{\text{A} }\) |
|
| \(\ds\tan(\theta) = \frac{\text{O} }{\text{A} }\) |
\(\ds\cot(\theta) = \frac{\text{A} }{\text{O} }\) |
Subsection B.3.2 Common Trigonometric Identities
\(\displaystyle \sin^2(x)+\cos^2(x)= 1\)
\(\displaystyle \tan^2(x)+ 1 = \sec^2(x)\)
\(\displaystyle 1 + \cot^2(x)=\csc^2(x)\)
\(\displaystyle \sin(2x) = 2\sin(x)\cos(x)\)
\begin{align*}
\cos(2x) \amp = \cos^2(x) - \sin^2(x) \amp \amp \\
\amp = 2\cos^2(x)-1 \amp \amp \\
\amp = 1-2\sin^2(x) \amp \amp
\end{align*}
\(\displaystyle \tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}\)
\(\displaystyle \sin\left(\frac{\pi}{2}-x\right) = \cos(x)\)
\(\displaystyle \cos\left(\frac{\pi}{2}-x\right) = \sin(x)\)
\(\displaystyle \tan\left(\frac{\pi}{2}-x\right) = \cot(x)\)
\(\displaystyle \csc\left(\frac{\pi}{2}-x\right) = \sec(x)\)
\(\displaystyle \sec\left(\frac{\pi}{2}-x\right) = \csc(x)\)
\(\displaystyle \cot\left(\frac{\pi}{2}-x\right) = \tan(x)\)
\(\displaystyle \sin(-x) = -\sin(x)\)
\(\displaystyle \cos (-x) = \cos(x)\)
\(\displaystyle \tan (-x) = -\tan(x)\)
\(\displaystyle \csc(-x) = -\csc(x)\)
\(\displaystyle \sec (-x) = \sec(x)\)
\(\displaystyle \cot (-x) = -\cot(x)\)
\(\displaystyle \sin^2(x) = \frac{1-\cos(2x)}{2}\)
\(\displaystyle \cos^2(x) = \frac{1+\cos(2x)}{2}\)
\(\displaystyle \tan^2(x) = \frac{1-\cos(2x)}{1+\cos(2x)}\)
\(\displaystyle \sin(x)+\sin(y) = 2\sin\left(\frac{x+y}2\right)\cos\left(\frac{x-y}2\right)\)
\(\displaystyle \sin(x)-\sin(y) = 2\sin\left(\frac{x-y}2\right)\cos\left(\frac{x+y}2\right)\)
\(\displaystyle \cos(x)+\cos(y) = 2\cos\left(\frac{x+y}2\right)\cos\left(\frac{x-y}2\right)\)
\(\displaystyle \cos(x)-\cos(y) = -2\sin\left(\frac{x+y}2\right)\sin\left(\frac{x-y}2\right)\)
\(\displaystyle \sin(x)\sin(y) = \frac12 \big(\cos(x-y) - \cos (x+y)\big)\)
\(\displaystyle \cos(x)\cos(y) = \frac12\big(\cos (x-y) +\cos (x+y)\big)\)
\(\displaystyle \sin(x)\cos(y) = \frac12 \big(\sin(x+y) + \sin (x-y)\big)\)
\(\displaystyle \sin (x\pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)\)
\(\displaystyle \cos (x\pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)\)
\(\displaystyle \tan (x\pm y) = \frac{\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}\)
