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Section B.3 Trigonometry Reference

The Unit Circle.

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

List B.3.1. Pythagorean Identities
  1. \(\displaystyle \sin ^2x+\cos ^2x= 1\)

  2. \(\displaystyle \tan^2x+ 1 = \sec^2 x\)

  3. \(\displaystyle 1 + \cot^2x=\csc^2 x\)

List B.3.2. Double Angle Formulas
  1. \(\displaystyle \sin 2x = 2\sin x\cos x\)

  2. \begin{align*} \cos 2x \amp = \cos^2x - \sin^2 x \amp \amp \\ \amp = 2\cos^2x-1 \amp \amp \\ \amp = 1-2\sin^2x \amp \amp \end{align*}
  3. \(\displaystyle \tan 2x = \frac{2\tan x}{1-\tan^2 x}\)

List B.3.3. Cofunction Identities
  1. \(\displaystyle \sin\left(\frac{\pi}{2}-x\right) = \cos x\)

  2. \(\displaystyle \cos\left(\frac{\pi}{2}-x\right) = \sin x\)

  3. \(\displaystyle \tan\left(\frac{\pi}{2}-x\right) = \cot x\)

  4. \(\displaystyle \csc\left(\frac{\pi}{2}-x\right) = \sec x\)

  5. \(\displaystyle \sec\left(\frac{\pi}{2}-x\right) = \csc x\)

  6. \(\displaystyle \cot\left(\frac{\pi}{2}-x\right) = \tan x\)

List B.3.4. Even/Odd Identities
  1. \(\displaystyle \sin(-x) = -\sin x\)

  2. \(\displaystyle \cos (-x) = \cos x\)

  3. \(\displaystyle \tan (-x) = -\tan x\)

  4. \(\displaystyle \csc(-x) = -\csc x\)

  5. \(\displaystyle \sec (-x) = \sec x\)

  6. \(\displaystyle \cot (-x) = -\cot x\)

List B.3.5. Power-Reducing Formulas
  1. \(\displaystyle \sin^2 x = \frac{1-\cos 2x}{2}\)

  2. \(\displaystyle \cos^2 x = \frac{1+\cos 2x}{2}\)

  3. \(\displaystyle \tan^2x = \frac{1-\cos 2x}{1+\cos 2x}\)

List B.3.6. Sum to Product Formulas
  1. \(\displaystyle \sin x+\sin y = 2\sin \left(\frac{x+y}2\right)\cos\left(\frac{x-y}2\right)\)

  2. \(\displaystyle \sin x-\sin y = 2\sin \left(\frac{x-y}2\right)\cos\left(\frac{x+y}2\right)\)

  3. \(\displaystyle \cos x+\cos y = 2\cos \left(\frac{x+y}2\right)\cos\left(\frac{x-y}2\right)\)

  4. \(\displaystyle \cos x-\cos y = -2\sin \left(\frac{x+y}2\right)\sin\left(\frac{x-y}2\right)\)

List B.3.7. Product to Sum Formulas
  1. \(\displaystyle \sin x\sin y = \frac12 \big(\cos(x-y) - \cos (x+y)\big)\)

  2. \(\displaystyle \cos x\cos y = \frac12\big(\cos (x-y) +\cos (x+y)\big)\)

  3. \(\displaystyle \sin x\cos y = \frac12 \big(\sin(x+y) + \sin (x-y)\big)\)

List B.3.8. Angle Sum/Difference Formulas
  1. \(\displaystyle \sin (x\pm y) = \sin x\cos y \pm \cos x\sin y\)

  2. \(\displaystyle \cos (x\pm y) = \cos x\cos y \mp \sin x\sin y\)

  3. \(\displaystyle \tan (x\pm y) = \frac{\tan x\pm \tan y}{1\mp \tan x\tan y}\)

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