Chapter 7: Analytic Trigonometry

In Problems 9–36, find the exact value of each expression.

$\tan \left[{\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$

Multiply:$\frac{\cos \left(\theta \right)}{1-\sin \left(\theta \right)}\frac{1+\sin \left(\theta \right)}{1+\sin \left(\theta \right)}$

In Problems 9–23, find the exact value, if any, of each composite function. If there is no value, say it is “not defined.” Do not use a calculator.

${\tan }^{-1}\left(\tan \frac{2\mathrm{\pi }}{3}\right)$

In Problems 11–34, solve each equation on the interval $0\le \theta \le 2\pi$.

$2\sin \theta +3=2$

For the function $f\left(x\right)=2x^5-x^4-4x^3+2x^2+2x-1$:

Part (a): Find the real zeros and their multiplicity.

Part (b): Find the intercepts.

Part (c): Find the power function that the graph of f resembles for large $\left|x\right|$.

Part (d): Graph fusing a graphing utility.

Part (e): Approximate the turning points, if any exist.

Part (f): Use the information obtained in parts (a)–(e) to sketch a graph of fby hand.

Part (g): Identify the intervals on which f is increasing, decreasing, or constant.

Establish the identity

$\frac{csc\theta +cot\theta }{sec\theta +\tan \theta }=\frac{sec\theta -\tan \theta }{csc\theta -cot\theta }$

True or False$\sin ({\sin }^{-1}0)=0and\cos ({\cos }^{-1}0)=0.$

Find the exact value of the expression:

$\sin \left(\frac{5\mathrm{\pi }}{12}\right)$

Use the information given about the angle $0<\theta <2\mathrm{\pi }$, to find the exact value of

$\cos \theta =\frac{-\sqrt{6}}{3}$