Exam 7: Trigonometric Identities and Equations

Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - $4 \sin ^ { 2 } x - 1 = 0$

Verify that each equation is an identity. - $\tan \theta \cdot \csc \theta = \sec \theta$

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\cos 2 x = 2 \cos ^ { 2 } x - 1$

Graph the inverse circular function. - $y=\operatorname{arccsc} \frac{1}{2} x$

Express the function as a trigonometric function of x. - $\tan 4 x$

Solve the equation for solutions over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate. - $\tan 2 x + \sec 2 x = 2$

Determine all solutions of the equation in radians. -Find $\cos \frac { \theta } { 2 }$ , given that $\cos \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $90 ^ { \circ } < \theta < 180 ^ { \circ }$ .

Use identities to write the expression as a single function of x or θ. - $\cos \left( \theta + 90 ^ { \circ } \right)$

Solve the problem. -A formula for the up-and-down motion of a weight on a spring is given by $S = 2 \sin \left( \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \right) \mathrm { t }$ , where $\mathrm { k }$ is the spring constant, $\mathrm { m }$ is the mass, and $\mathrm { t }$ is the time. Solve the equation for $\mathrm { t }$ .

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\tan ^ { 2 } \theta \csc ^ { 2 } \theta$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { \sec \theta \sin \theta } { \tan \theta } - 1$

Use a sum or difference identity to find the exact value. - $\sin \frac { 7 \pi } { 24 } \cos \frac { \pi } { 8 } - \cos \frac { 7 \pi } { 24 } \sin \frac { \pi } { 8 }$

Give the degree measure of θ. - $\theta = \arctan \left( \frac { \sqrt { 3 } } { 3 } \right)$

Solve the equation for exact solutions over the interval [0, 2π). - $\sin ^ { 2 } x - \cos ^ { 2 } x = 0$

Use a sum or difference identity to find the exact value. - $\sin 15 ^ { \circ }$

Use the cofunction identities to find an angle θ that makes the statement true. - $\sin \left( 3 \theta - 17 ^ { \circ } \right) = \cos \left( \theta + 43 ^ { \circ } \right)$

Use a sum or difference identity to find the exact value. - $\tan \left( - \frac { 5 \pi } { 12 } \right)$

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $( \csc \theta + \cot \theta ) ( 1 - \cos \theta ) = \sin \theta$