91Ó°ÊÓ

What is the angle between the positive horizontal axis and the line containing the points (2,5) and (6,2)\(?\)

Find an identity expressing \(\tan \left(\sin ^{-1} t\right)\) as a nice function of \(t\).

Suppose \(f\) is a function with period \(p\). Explain why \(f(x+2 p)=f(x)\) for every number \(x\) in the domain of \(f\).

Find a nice formula for \(\sin (5 \theta)\) in terms of \(\sin \theta\).

Evaluate the given quantities assuming that \(u\) and \(v\) are both in the interval \(\left(-\frac{\pi}{2}, 0\right)\) and $$ \tan u=-\frac{1}{7} \quad \text { and } \quad \tan v=-\frac{1}{8}. $$ \(\sin (2 u)\)

For Exercises 49-52: Hilly areas often have road signs giving the percentage grade for the road. A 5\% grade, for example, means that the altitude changes by 5 feet for each 100 feet of horizontal distance. What percentage grade should be put on a road sign where the angle of elevation of the road is \(3^{\circ} ?\)

Show that $$ \sin u \sin v=\frac{\cos (u-v)-\cos (u+v)}{2} $$ for all \(u, v\).

Suppose \(f\) is a function with period \(p\). Explain why \(f(x-p)=f(x)\) for every number \(x\) such that \(x-p\) is in the domain of \(f\).

Show that $$ \cos \left(\tan ^{-1} t\right)=\frac{1}{\sqrt{1+t^{2}}} $$ for every number \(t\).

Evaluate the given quantities assuming that \(u\) and \(v\) are both in the interval \(\left(-\frac{\pi}{2}, 0\right)\) and $$ \tan u=-\frac{1}{7} \quad \text { and } \quad \tan v=-\frac{1}{8}. $$ \(\sin (2 v)\)