Exam 9: Trigonometric Identities and Equations

Solve each equation or inequality over the indicated interval. - $2 \sin ^{2} x-3 \cos x-1, \quad[0,2 \pi)$

Given $\sin y=-\frac{4}{5}, \cos x=\frac{1}{2}$ with $\frac{3 \pi}{2}<x<2 \pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following: - $\cos (x+y)$

Suppose that the formula $A(t)=\frac{1}{3} \sin \frac{\pi}{4} t+\frac{2}{3}$ describes the motion formed by a rhythmically moving arm during an 8-minute time period where $A(t)$ is the angle (in radians) formed by the arm at time $t$ (in minutes). (a) Give the domain and range of $A$ . (b) Graph $A(t)$ over its domain. (c) Use the graph to determine the maximum and minimum values of $A(t)$ and when they occur. (d) Find $A(1)$ analytically and check your result graphically. Use symmetry to find $A(3)$ . (e) When is the angle $\frac{2}{3}$ radians? (f) Write the equation $A=\frac{1}{3} \sin \frac{\pi}{4} t+\frac{2}{3}$ as an equation involving arcsine by solving for $t$ .

Graph $\sec x-\tan x$ , and use the graph to conjecture an identity. Verify your conjecture analytically.

Find the exact value of each expression. - $\sin \left(2 \arccos \frac{3}{7}\right)$

Given $\sin y=-\frac{2}{5}, \cos x=-\frac{1}{3}$ with $\frac{\pi}{2}<x<\pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following: - $\cos 2 x$

Graph $\csc x+\cot x$ , and use the graph to conjecture an identity. Verify your conjecture analytically.

Find the exact value of each expression. - $\sin \left(\cos ^{-1} \frac{2}{3}\right)$

Verify that each equation is an identity. - $\frac{2 \tan x}{1+\tan ^{2} x}=\sin 2 x$

Graph $\tan x+\sec x$ , and use the graph to conjecture an identity. Verify your conjecture analytically.

Verify that each equation is an identity. - $\frac{\sec ^{2} x-\tan ^{2} x}{\cos ^{2} x}-1=\tan ^{2} x$

Write as an algebraic expression in $u, u>0$ . - $\cos (\arctan u)$

Solve each equation or inequality over the indicated interval. - $1+\sqrt{2} \sin x \geq 0, \quad[0,2 \pi)$

Use an identity to write each expression as a trigonometric function of $\theta$ alone. - $\sin (\theta-\pi)$

Use an identity to write each expression as a trigonometric function of $\theta$ alone. - $\cos \left(270^{\circ}+\theta\right)$

Solve each equation or inequality over the indicated interval. - $\cos 2 \theta-\cos \theta=0, \quad\left[0^{\circ}, 360^{\circ}\right)$

Find the exact value of each expression. - $\tan ^{-1}(\sqrt{3})$

Solve each equation or inequality over the indicated interval. - $3 \tan ^{2} \theta-1=0,\left[0^{\circ}, 360^{\circ}\right)$

Express $\cot ^{2} x-\csc ^{2} x$ in terms of $\sin x$ and $\cos x$ , and simplify.

Suppose that the formula $A(t)=-\frac{1}{4} \cos \frac{\pi}{8} t+\frac{1}{4}$ describes the motion formed by a rhythmically moving arm during a 16 minute time period where $A(t)$ is the angle (in radians) formed by the arm at time $t$ (in minutes). (a) Give the domain and range of $A$ . (b) Graph $A(t)$ over its domain. (c) Use the graph to determine the maximum and minimum values of $A(t)$ and when they occur. (d) Find $A(6)$ analytically and check your result graphically. Use symmetry to find $A(10)$ . (e) When is the angle $\frac{1}{4}$ radians? (f) Write the equation $A=-\frac{1}{4} \cos \frac{\pi}{8} t+\frac{1}{4}$ as an equation involving arcsine by solving for $t$ .