Chapter 7: Analytic Trigonometry

If$f\left(x\right)=2x^2+3x+1$and $g\left(x\right)=x^2+3x+2$, solve:

Part (a): $f\left(x\right)=0$

Part (b):$f\left(x\right)=g\left(x\right)$

Part (c): $f\left(x\right)>0$

Part (d):$f\left(x\right)\ge g\left(x\right)$

Establish the identity

$\sin \theta \tan \theta +\cos \theta =sec\theta$

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.

${\cos }^{-1}\left(\cos \frac{15\mathrm{\pi }}{7}\right)$

Find the exact value of the expression:

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

Express the product $\sin 4\theta \cos 6\theta$as a sum containing only sines or only cosines.

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

$a.\sin 2\theta b.\cos 2\theta c.\sin \frac{\theta }{2}d.\cos \frac{\theta }{2}e.\tan \left(2\theta \right)f.\tan \frac{\theta }{2}$

$\sin \theta =-\frac{\sqrt{3}}{3},\frac{3\pi }{2}<\theta <2\mathrm{\pi }$

Provide a justification as to why no further points of intersection (and therefore solutions) exist in Figure 30 for $x<-\mathrm{\pi }$or $x>4\mathrm{\pi }$.

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

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

Rewrite over a common denominator:

$\frac{\sin \left(\theta \right)+\cos \left(\theta \right)}{\cos \left(\theta \right)}+\frac{\sin \left(\theta \right)+\cos \left(\theta \right)}{\sin \left(\theta \right)}$

In Problems 13–24, find the exact value of each expression.

sin^-1 0