91Ó°ÊÓ

Convert each of the following degree measurements to radians: a. \(50^{\circ}\) b. \(120^{\circ}\) c. \(-15^{\circ}\)

Convert each of the following radian measurements to degrees: a. \(0.25\) radian b. 1 radian c. \(-1.5\) radians

Find \(\tan \theta\) if \(\sin \theta=\frac{4}{5}\) and \(0 \leq \theta \leq \frac{\pi}{2}\).

Find \(\csc \theta\) if \(\cot \theta=\frac{\sqrt{5}}{2}\) and \(0 \leq \theta \leq \frac{\pi}{2}\).

Starting with the addition formulas for the sine and cosine, derive these identities: \(\cos \left(\frac{\pi}{2}+\theta\right)=-\sin \theta \quad\) and \(\quad \sin \left(\frac{\pi}{2}+\theta\right)=\cos \theta\) Give geometric arguments to justify the identities.

Use the addition formulas for sine and cosine to derive the double-angle formulas $$ \begin{aligned} \sin (2 A) &=2 \sin A \cos A \\ \cos (2 A) &=\cos ^{2} A-\sin ^{2} A \\ &=2 \cos ^{2} A-1 \\ &=1-2 \sin ^{2} A \end{aligned} $$

a. Use the double-angle formulas along with the Pythagorean identity \(\sin ^{2} A+\cos ^{2} A=1\) to show that \(\cos ^{2} \theta=\frac{1}{2}(1+\cos 2 \theta) \quad\) and \(\sin ^{2} \theta=\frac{1}{2}(1-\cos 2 \theta)\) b. Use the identities in part (a) to show that $$ \begin{aligned} &\int \cos ^{2} x d x=\frac{1}{2} x+\frac{1}{4} \sin (2 x)+C \\ &\int \sin ^{2} x d x=\frac{1}{2} x-\frac{1}{4} \sin (2 x)+C \end{aligned} $$ and c. An object moves along a straight line in such a way that after \(t\) seconds, its velocity is given by $$ v(t)=2 t+\sin ^{2}\left(\frac{\pi t}{6}\right) $$ meters per second. Find the average velocity of the object over the time period \(0 \leq t \leq 3\).

Differentiate the given function. $$f(x)=\cos (1-5 x)$$

Differentiate the given function. $$f(x)=\sin (3 x+1) \cos x$$

Differentiate the given function. $$f(x)=\cos ^{2} x$$