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All my math notes, now in Markdown.

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Fundamental Trigonometric Identities

Learning Targets

You should be able to

Concepts / Definitions

Reciprocal Identities

$\csc\theta = \frac{1}{\sin\theta}$ $\qquad$ $\sec\theta = \frac{1}{\cos\theta}$ $\qquad$ $\cot\theta = \frac{1}{\tan\theta}$
$\sin\theta = \frac{1}{\csc\theta}$ $\qquad$ $\cos\theta = \frac{1}{\sec\theta}$ $\qquad$ $\tan\theta = \frac{1}{\cot\theta}$

Quotient Identities

$\tan\theta = \frac{\sin\theta}{\cos\theta}$ $\qquad\qquad$ $\cot\theta = \frac{\cos\theta}{\sin\theta}$

Odd-Even Identities

$\sin(-\theta) = -\sin\theta$ $\qquad$ $\csc(-\theta) = -\csc\theta$
$\cos(-\theta) = \cos\theta$ $\qquad$ $\sec(-\theta) = \sec\theta$
$\tan(-\theta) = -\tan\theta$ $\qquad$ $\cot(-\theta) = -\cot\theta$

Cofunction Identities

(co is for complement)
$\sin(\frac \pi2-\theta) = \cos\theta$ $\qquad$ $\cos(\frac \pi2-\theta) = \sin\theta$
$\tan(\frac \pi2-\theta) = \cot\theta$ $\qquad$ $\cot(\frac \pi2-\theta) = \tan\theta$
$\sec(\frac \pi2-\theta) = \csc\theta$ $\qquad$ $\csc(\frac \pi2-\theta) = \sec\theta$

Pythagorean Identites

Exercises

Simplify the following

  1. $\tan(\theta)\csc(\theta)$
  2. $\cot^2x-\csc^2x$
  3. $\frac{\sin(\frac \pi2 - \theta)}{\cos(\frac \pi2 - \theta)}$
  4. $\frac{\sin^2x}{1+\cos x}$
  5. $\sec^4y-\tan^4y$
  6. $\frac{\sec^2x-1}{\sin^2x}$
  7. $\frac{1+\tan x}{1+\cot x}$
  8. $\frac{\sec^2x\csc x}{\sec^2x+\csc^2x}$
  9. $\frac{1}{1-\sin\varphi} + \frac{1}{1+\sin\varphi}$
  10. $\frac{\sin x}{\cot^2 x} - \frac{\sin x}{\cos^2 x}$

Factor the following

  1. $\sin^2\theta + \frac{2}{\csc\theta} + 1$
  2. $\sec^2x - \sec x + \tan^2 x$
  3. Evaluate without calculator using identities $\cos(-\theta),\ if\ \sin(\theta-\frac \pi2) = 0.73$