Exam 7: Analytic Trigonometry

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

Use addition or subtraction formulas to find the exact value of $\tan \left( - 15 ^ { \circ } \right)$ .

Use an addition or subtraction formula to write $\frac { \tan \frac { 2 \pi } { 15 } + \tan \frac { 6 \pi } { 5 } } { 1 - \tan \frac { 2 \pi } { 15 } \tan \frac { 6 \pi } { 5 } }$ and find the exact value.

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

Find all solutions of $4 \sin ^ { 2 } x - 2 ( \sqrt { 3 } + \sqrt { 2 } ) \sin x + \sqrt { 6 } = 0$ .

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

Simplify. $\frac { \sin x \cos x } { \tan x }$

Use a half-angle formula to find the exact value of $\sin \frac { \pi } { 8 }$ .

Prove $\frac { \sin 4 x + \sin 2 x } { \cos 4 x + \cos 2 x } = \tan 3 x$ .

Sketch the graphs of $f ( x ) = 2 \sin x$ and $g ( x ) = \sin 2 x$ on the same axes and find their points of intersection.

Use a double-angle formula or half-angle formula to simplify $\frac { \sin 48 ^ { \circ } } { 1 + \cos 48 ^ { \circ } }$ .

Find all solutions of $\tan ^ { 4 } x \sin x - 9 \sin x = 0$ .

Verify the identity. $\sin ( - x ) \tan ( - x ) = \sin ^ { 2 } x \sec x$

Find all solutions of $\cos ^ { 2 } x = \cos x + 6$ .

Find $\sin 2 x$ , $\cos 2 x$ and $\tan 2 x$ , given that $\tan \theta = \frac { 1 } { 2 }$ and $\sin \theta < 0$ .

Use addition or subtraction formulas to find the exact value of $\tan \frac { 7 \pi } { 12 }$ .

Verify the identity. $( \tan x + \cot x ) ^ { 2 } = \sec ^ { 2 } x + \csc ^ { 2 } x$

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

Solve the given equation. $\sin \theta = \frac { \sqrt { 2 } } { 2 }$

Find the exact value of the expression. $\cos \left( \sin ^ { - 1 } ( \sqrt { 3 } / 2 ) + \tan ^ { - 1 } ( \sqrt { 3 } ) \right)$