Exam 7: Analytic Trigonometry

Solve the given equation. $2 \cos ^ { 2 } \theta + 5 \cos \theta - 12 = 0$

Find all solutions of $\cos ^ { 2 } x = 1 - 2 \sin ^ { 2 } x$ .

Find $\sin 2 x$ , $\cos 2 x$ and $\tan 2 x$ , given that $\cos x = \frac { 5 } { 13 }$ and $x$ is in quadrant IV.

Find $\sin 2 x$ , $\cos 2 x$ and $\tan 2 x$ , given that $\cot x = - \frac { 3 } { 4 }$ and $\sin x < 0$ .

Use a graphing device to find the solution of $16 e ^ { \cos x } = x ^ { 2 }$ , correct to one decimal place.

Write $\frac { \sec x - \cos x } { \tan x }$ in terms of $\sin x$ and $\cos x$ then simplify.

Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that $0 \leq x \leq \pi / 2$ $\frac { \sqrt { x ^ { 2 } - 36 } } { x } , x = 6 \sec \theta$

Find all solutions of the given equation. $5 \cos \theta - 1 = 0$

Find the exact value of the expression. $\tan \left( \sin ^ { - 1 } \frac { 1 } { 3 } - \cos ^ { - 1 } \frac { 3 } { 4 } \right)$

Write the given expression in terms of x and y only. $\sin \left( \sin ^ { - 1 } y + \cos ^ { - 1 } x \right)$

Write the product as a sum. $\cos 7 x \cos 3 x$

Use an addition or subtraction formula to write $\frac { \tan 10 ^ { \circ } + \tan 20 ^ { \circ } } { 1 - \tan 10 ^ { \circ } \tan 20 ^ { \circ } }$ and find its exact value.

Simplify. $\frac { \sin x } { \csc x + \cot x }$

Use a half-angle formula to find the exact value of $\cos \left( - 15 ^ { \circ } \right)$ .

Verify the identity. $\frac { 1 - \sin t } { \cos t } + \frac { \cos t } { 1 - \sin t } = 2 \operatorname { sect }$

Find all solutions of $\tan \frac { x } { 3 } - 1 = 0$ .

Use an addition or subtraction formula to write $\cos 75 ^ { \circ } \cos 15 ^ { \circ } + \sin 75 ^ { \circ } \sin 15 ^ { \circ }$ and find its exact value.

Simplify. $\frac { 1 + \cos x } { 1 + \sec x }$

Solve the given equation. $\sec ^ { 2 } \theta - 4 = 0$

Use an addition or subtraction formula to write $\sin 74 ^ { \circ } \cos 14 ^ { \circ } - \cos 74 ^ { \circ } \sin 14 ^ { \circ }$ and find its exact value.