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Subsection A.1 Trigonometry
The Unit Circle.
Unit Circle Definitions of the Trigonometric Functions.
| \(\sin \theta = y\) |
\(\cos \theta = x\) |
| \(\displaystyle \csc \theta = \frac{1}{y}\) |
\(\displaystyle \sec \theta = \frac{1}{x}\) |
| \(\displaystyle \tan \theta = \frac{y}{x}\) |
\(\displaystyle \cot \theta = \frac{x}{y}\) |
Table A.1.14. Right Triangle Definition of the Trigonometric Functions.
| \(\displaystyle \sin \theta = \frac{\text{O}}{\text{H}}\) |
\(\displaystyle \csc \theta = \frac{\text{H}}{\text{O}}\) |
|
| \(\displaystyle \cos \theta = \frac{\text{A}}{\text{H}}\) |
\(\displaystyle \sec \theta = \frac{\text{H}}{\text{A}}\) |
|
| \(\displaystyle \tan \theta = \frac{\text{O}}{\text{A}}\) |
\(\displaystyle \cot \theta = \frac{\text{A}}{\text{O}}\) |
Table A.1.15.
Subsubsection A.1.1 Pythagorean Identities
\(\displaystyle \displaystyle \sin ^2x+\cos ^2x= 1\)
\(\displaystyle \displaystyle \tan ^2x+ 1 = \sec ^2 x\)
\(\displaystyle \displaystyle 1 + \cot ^2x=\csc ^2 x\)
Subsubsection A.1.2 Cofunction Identities
\(\displaystyle \displaystyle \sin \left(\frac{\pi }{2}-x\right) = \cos x\)
\(\displaystyle \displaystyle \cos \left(\frac{\pi }{2}-x\right) = \sin x\)
\(\displaystyle \displaystyle \tan \left(\frac{\pi }{2}-x\right) = \cot x\)
\(\displaystyle \displaystyle \csc \left(\frac{\pi }{2}-x\right) = \sec x\)
\(\displaystyle \displaystyle \sec \left(\frac{\pi }{2}-x\right) = \csc x\)
\(\displaystyle \displaystyle \cot \left(\frac{\pi }{2}-x\right) = \tan x\)
Subsubsection A.1.3 Double Angle Formulas
\(\displaystyle \sin 2x = 2\sin x\cos x\)
\(\displaystyle \begin{array}{l}
\cos 2x = \cos ^2x - \sin ^2 x \\
\phantom{\cos 2x}= 2\cos ^2x-1 \\
\phantom{\cos 2x}= 1-2\sin ^2x
\end{array}\)
\(\displaystyle \displaystyle \tan 2x = \frac{2\tan x}{1-\tan ^2 x}\)
Subsubsection A.1.4 Sum to Product Formulas
\(\displaystyle \displaystyle \sin x+\sin y = 2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)\)
\(\displaystyle \displaystyle \sin x-\sin y = 2\sin \left(\frac{x-y}{2}\right)\cos \left(\frac{x+y}{2}\right)\)
\(\displaystyle \displaystyle \cos x+\cos y = 2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)\)
\(\displaystyle \displaystyle \cos x-\cos y = -2\sin \left(\frac{x+y}{2}\right)\sin \left(\frac{x-y}{2}\right)\)
Subsubsection A.1.5 Power–Reducing Formulas
\(\displaystyle \displaystyle \sin ^2 x = \frac{1-\cos 2x}{2}\)
\(\displaystyle \displaystyle \cos ^2 x = \frac{1+\cos 2x}{2}\)
\(\displaystyle \displaystyle \tan ^2x = \frac{1-\cos 2x}{1+\cos 2x}\)
Subsubsection A.1.6 Even/Odd Identities
\(\displaystyle \sin (-x) = -\sin x\)
\(\displaystyle \cos (-x) = \cos x\)
\(\displaystyle \tan (-x) = -\tan x\)
\(\displaystyle \csc (-x) = -\csc x\)
\(\displaystyle \sec (-x) = \sec x\)
\(\displaystyle \cot (-x) = -\cot x\)
Subsubsection A.1.7 Product to Sum Formulas
\(\displaystyle \displaystyle \sin x\sin y = \frac{1}{2} \big (\cos (x-y) - \cos (x+y)\big )\)
\(\displaystyle \displaystyle \cos x\cos y = \frac{1}{2}\big (\cos (x-y) +\cos (x+y)\big )\)
\(\displaystyle \displaystyle \sin x\cos y = \frac{1}{2} \big (\sin (x+y) + \sin (x-y)\big )\)
Subsubsection A.1.8 Angle Sum/Difference Formulas
\(\displaystyle \sin (x\pm y) = \sin x\cos y \pm \cos x\sin y\)
\(\displaystyle \cos (x\pm y) = \cos x\cos y \mp \sin x\sin y\)
\(\displaystyle \displaystyle \tan (x\pm y) = \frac{\tan x\pm \tan y}{1\mp \tan x\tan y}\)
