Chapter 7: Analytic Trigonometry

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

$\sin \left({\cos }^{-1}\frac{1}{2}\right)$

Rewrite in terms of sine and cosine functions: $cot\left(\theta \right)sec\left(\theta \right)$

True or False$Thedomainofy={\sin }^{-1}xis-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2}.$

True or FalseIf$f\left(x\right)=\sin x$ and$g\left(x\right)=\cos x$ then $g(\alpha +\beta )=g\left(\alpha \right)g\left(\beta \right)-f\left(\alpha \right)f\left(\beta \right)$.

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.

${\cos }^{-1}\left(\cos \frac{3\mathrm{\pi }}{4}\right)$

True or False The equation $\sin \theta =2$has a real solution that can be found using a calculator.

Express the product $\sin \left(3\theta \right)\sin \left(5\theta \right)$as a sum containing only sines or only cosines.

Use a calculator to evaluate the expression$co{t}^{-1}5$. Express angles in radians rounded to two decimal places.

If $\sin \alpha =\frac{1}{3},\frac{\mathrm{\pi }}{2}<\alpha <\mathrm{\pi }$and $\cos \beta =-\frac{1}{3},\mathrm{\pi }<\mathrm{\beta }<\frac{3\mathrm{\pi }}{2}$, find the exact value of:

Part (a):$\cos \alpha$

Part (b): $\sin \beta$

Part (c):$\cos \left(2\alpha \right)$

Part (d):$\cos \left(\alpha +\beta \right)$

Part (e):$\sin \frac{\beta }{2}$

Use the information given about the angle , to find the exact value of

$\tan \theta =\frac{1}{2},\pi <\theta <\frac{3\pi }{2}$