Exam 5: Trigonometric Identities

Use an identity to write the expression as a single trigonometric function or as a single number. -Use the exact value $\cos 72 ^ { \circ } = \frac { \sqrt { 5 } - 1 } { 4 }$ and identities to find the exact value of $\sin 72 ^ { \circ }$ .

Solve the problem. -Find $\sec \theta$ if $\cos \theta = \frac { 1 } { 4 }$ and $\sin \theta > 0$ .

Verify that the equation is an identity. - $\sin ( \alpha - \beta ) \cos ( \alpha + \beta ) = \sin \alpha \cos \alpha - \sin \beta \cos \beta$

Use a sum or difference identity to find the exact value. -

Solve the problem. - $\sqrt { \frac { 1 + \cos 18 ^ { \circ } } { 1 - \cos 18 ^ { \circ } } }$

Use the cofunction identities to find an angle that makes the statement true. -Find $\tan \frac { x } { 2 }$ , given that $\tan x = 3$ and $x$ terminates in $\pi < x < \frac { 3 \pi } { 2 }$ .

Determine all solutions of the equation in radians. -Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 4 } { 5 }$ , with $\mathrm { s }$ in quadrant II, and $\cos \mathrm { t } = \frac { 12 } { 13 }$ , with $\mathrm { t }$ in quadrant IV.

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. -sin 15°

Write in terms of the cofunction of a complementary angle. Find the exact value of the expression using the provided information. - $\frac { \tan \theta } { \sec \theta }$

Perform the indicated operations and simplify the result so there are no quotients. - $\tan \theta = \frac { 12 } { 5 } , \pi < \theta < \frac { 3 \pi } { 2 } \quad$ Find $\sin ( 2 \theta )$ .

Verify that each equation is an identity. - $\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)$

Find the exact value of the expression using the provided information. -An airplane flying faster than the speed of sound creates sound waves that form a cone. If vertex angle of the cone and $\mathrm { m }$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \mathrm { m }$ $( \mathrm { m } > 1 )$ . Find the Mach number to the nearest tenth if $\theta = 30 ^ { \circ }$ .

Use the cofunction identities to find an angle that makes the statement true. -sin (ϴ + 4°)= cos (ϴ + 6°) A

Using a sum or difference identity, write the following as an expression involving functions of x. - $\sin \left( x - \frac { \pi } { 2 } \right)$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { \tan \theta } { \cot \theta }$

Using a sum or difference identity, write the following as an expression involving functions of x. - $\tan ( x + \pi )$

Verify that each equation is an identity. - $\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )$

Use identities to find the indicated value for each angle measure. - $\cos x ( \csc x - \sec x ) - \cot x$

Factor the trigonometric expression and simplify. - $\tan \left( x - \frac { \pi } { 3 } \right)$

Verify that each equation is an identity. - $\csc ^ { 2 } t - \cos t \sec t = \cot ^ { 2 } t$