91Ó°ÊÓ

Suppose \(u\) and \(v\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) with \(\tan u=2\) and \(\tan v=3\). Find exact expressions for the indicated quantities. $$ \sec u $$

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 4 and 5 in Section 6.3.] $$ \sin \frac{5 \pi}{12} $$

(a) Sketch a radius of the unit circle making an angle \(\theta\) with the positive horizontal axis such that \(\cos \theta=\frac{6}{7}\). (b) Sketch another radius, different from the one in part (a), also illustrating \(\cos \theta=\frac{6}{7}\).

Suppose \(u\) and \(v\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) with \(\tan u=2\) and \(\tan v=3\). Find exact expressions for the indicated quantities. $$ \sec v $$

What is the slope of the radius of the unit circle that has a \(60^{\circ}\) angle with the positive horizontal axis?

(a) Sketch a radius of the unit circle making an angle \(\theta\) with the positive horizontal axis such that \(\sin \theta=-0.8\). (b) Sketch another radius, different from the one in part (a), also illustrating \(\sin \theta=\) -0.8.

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 4 and 5 in Section 6.3.] $$ \sin \frac{3 \pi}{8} $$

Explain why $$ \cos ^{-1} t=\sin ^{-1} \sqrt{1-t^{2}} $$ whenever \(0 \leq t \leq 1\)

Find the endpoint of the radius of the unit circle that makes the given angle with the positive horizontal axis. \(\frac{5 \pi}{2}\) radians

Explain why $$ \cos ^{-1} t=\tan ^{-1} \frac{\sqrt{1-t^{2}}}{t} $$ whenever \(0