Exam 8: Trigonometric Identities and Equations

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

Use identities to find the indicated value for each angle measure. - $\cos 2 \theta = \frac { 4 } { 5 }$ and $\theta$ terminates in quadrant III $\quad$ Find $\cos \theta$ .

Provide an appropriate response. -Graph the expression $\cos x + \sin x \tan x$ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

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

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

Use the cofunction identities to find an angle $\theta$ that makes the statement true. - $\cot \left( 2 \theta - 20 ^ { \circ } \right) = \tan \left( \theta - 10 ^ { \circ } \right)$

Use the fundamental identities to simplify the expression. - $\cos x \tan x$

Use a sum or difference identity to find the exact value. - $\sin \frac { 7 \pi } { 12 }$

Decide whether the expression is or is not an identity. - $\sqrt { \sin ^ { 2 } x } = \sin x$

Solve the problem. - $\tan ^ { 2 } \mathrm { u } ( 1 + \cos ( 2 \mathrm { u } ) ) = 1 - \cos ( 2 \mathrm { u } )$

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

Verify that each equation is an identity. - $\frac { 1 - \sec \theta } { \tan \theta } + \frac { \tan \theta } { 1 - \sec \theta } = - 2 \csc \theta$

Verify that each equation is an identity. - $\sec ^ { 4 } x - \tan ^ { 4 } x = \sec ^ { 2 } x + \tan ^ { 2 } x$

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\sin 2 x = 2 \sin x$

Write the following as an algebraic expression in u, u > 0. - $\sin \left( \arcsin \frac { \mathrm { u } } { \sqrt { 3 } } \right)$

Solve the equation for x, where x is restricted to the given interval. - $y = 3 \tan ( 2 x - 1 )$ , for $x$ in $\left( \frac { 1 } { 2 } - \frac { \pi } { 4 } , \frac { 1 } { 2 } + \frac { \pi } { 4 } \right)$

Use identities to find the indicated value for each angle measure. - $\cos 2 \theta = - \frac { 5 } { 7 }$ and $\frac { \pi } { 2 } < \theta < \pi \quad$ Find $\sin \theta$

Provide an appropriate response. -The statement s $\sin \left( \sin ^ { - 1 } x \right) = x$ x for all real numbers in the interval $0 \leq \mathrm { x } \leq \pi$

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. -Snell's Law states that $\frac { \sin \theta _ { 1 } } { \sin \theta _ { 2 } } = \frac { \mathrm { v } _ { 1 } } { \mathrm { v } _ { 2 } }$ where $\mathrm { v } _ { 1 } , \mathrm { v } _ { 2 }$ are the speeds at which light travels through two different mediums, and $\theta _ { 1 } , \theta _ { 2 }$ are the angles of incidence and refraction respectively. Find the angle of refraction if $\frac { { } ^ { v _ { 1 } } } { \mathrm { v } _ { 2 } } = 1.7$ and the angle of incidence is $14 ^ { \circ }$ . Give your answer in degrees to the nearest hundredth.

Use a sum or difference identity to find the exact value. - $\tan 105 ^ { \circ }$