Exam 7: Analytic Trigonometry

Find $\sin 2 x$ , $\cos 2 x$ and $\tan 2 x$ , given that $\sec x = - 4$ and $\sin x < 0$ .

Solve the given equation. $\tan \theta \sin \theta = - \sin \theta$

Prove $\sin ( \pi - x ) + \cos ( x - x ) = \sin x - \cos x$ .

Find the exact value. $\cos \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)$

Use a half-angle formula to find the exact value of $\tan \frac { 11 \pi } { 12 }$ .

Solve the given equation. $2 \sin ^ { 2 } \theta - 7 \sin \theta + 3 = 0$

Write $\cos x \sin x \sec x$ in terms of $\sin x$ and $\cos x$ then simplify.

Use the formula $\tan \psi = \left| \frac { m _ { 2 } - m _ { 1 } } { 1 + m _ { 1 } m _ { 2 } } \right|$ to find the acute angle $\psi$ formed by the two lines $y - x = 0$ and $2 x - 3 y = 1$ .

Prove $\cos \left( x - \frac { \pi } { 2 } \right) + \cos \left( x + \frac { \pi } { 2 } \right) = 0$ .

Simplify. $\frac { 1 + \sec x } { \sin x + \tan x }$

Solve the given equation. $\tan \theta = 4$

Find all solutions of $4 \sin ^ { 2 } x - 1 = 0$ .

Use an addition or subtraction formula to write $\cos 12 ^ { \circ } \cos 18 ^ { \circ } - \sin 12 ^ { \circ } \sin 18 ^ { \circ }$ and find its exact value.

Solve the given equation. $9 \cos ^ { 2 } \theta - 6 \cos \theta + 1 = 0$

Verify the identity. $\tan x \cos x = \sin x$

Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume that $0 \leq x \leq \pi / 2$ $\sqrt { 16 - x ^ { 2 } } , \quad x = 4 \sin \theta$

Prove $\sin ( x + y ) + \sin ( x - y ) = 2 \sin x \cos y$ .

Find all solutions of $2 \cos 3 x - 1 = 0$ .

Prove $\cos \left( x + \frac { \pi } { 4 } \right) + \cos \left( x - \frac { \pi } { 4 } \right) = \sqrt { 2 } \cos x$ .

Write the product as a sum. $\cos x \sin 2 x$