Exam 6: Analytic Trigonometry

Find the exact value under the given conditions. - $\sin \alpha = - \frac { 3 } { 5 } , \frac { 3 \pi } { 2 } < \alpha < 2 \pi ; \quad \cos \beta = - \frac { \sqrt { 21 } } { 5 } , \pi < \beta < \frac { 3 \pi } { 2 } \quad$ Find $\sin ( \alpha - \beta )$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\cos 22.5 ^ { \circ }$

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

Solve the problem. -If $\cos 2 \theta = - \frac { 24 } { 25 }$ , and $\frac { \pi } { 2 } < 2 \theta < \pi$ , then find $\sin \theta$

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\cos \theta = \frac { 4 } { 5 } , \frac { 3 \pi } { 2 } \leq \theta \leq 2 \pi$ Find $\cos \frac { \theta } { 2 }$ .

Complete the identity. - $\frac { \sin ( 4 \theta ) + \sin ( 8 \theta ) } { \cos ( 4 \theta ) + \cos ( 8 \theta ) } = ?$

Use the information given about the angle $\theta , 0 \leq \theta \leq 2 \pi$ , to find the exact value of the indicated trigonometric function. - $\csc \theta = \frac { 17 } { 15 } , \frac { \pi } { 2 } < \theta < \pi$ Find $\cos ( 2 \theta )$ .

Find the exact value under the given conditions. - $\cos \alpha = - \frac { 5 } { 13 } , \frac { \pi } { 2 } < \alpha < \pi ; \quad \sin \beta = \frac { 15 } { 17 } , \frac { \pi } { 2 } < \alpha < \pi \quad$ Find $\tan ( \alpha + \beta )$ .

Use a calculator to find the value of the expression in radian measure rounded to two decimal places. - $\cot ^ { - 1 } \left( - \frac { 5 } { 7 } \right)$

Solve the problem. -Rewrite in terms of sine and cosine: $\tan x \cdot \cot x$

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ . - $f ( x ) = 3 \sin x - 5$

Solve the problem. - $\sin 4 x = ( 4 \sin x \cos x ) \left( 1 - 2 \sin ^ { 2 } x \right)$

Solve the equation on the interval $0 \leq \theta < 2 \pi \text {. }$ - $\cos \theta + \sin \theta = 0$

Complete the identity. - $\sin ^ { 2 } \theta + \sin ^ { 2 } \theta \cot ^ { 2 } \theta =$ ?

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\csc \theta = - \frac { 3 } { 2 } , \tan \theta > 0 \quad$ Find $\cos \frac { \theta } { 2 }$ .

Find the exact solution of the equation. - $\cos ^ { - 1 } x = 0$

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

Find the exact value of the expression. - $\cos ^ { - 1 } [ \cos ( - 0.9372 ) ]$

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

Use the information given about the angle $\theta , 0 \leq \theta \leq 2 \pi$ , to find the exact value of the indicated trigonometric function. - $\csc \theta = - \frac { 3 } { 2 } , \quad \tan \theta > 0 \quad$ Find $\cos ( 2 \theta )$