Exam 8: Trigonometric Identities and Equations

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\tan ^ { 2 } 2 \theta = 5$

Use the fundamental identities to simplify the expression. - $\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)$

Perform the indicated operations and simplify the result so there are no quotients. - $2 \tan \theta - ( 1 + \tan \theta ) ^ { 2 }$

Perform the indicated operations and simplify the result so there are no quotients. - $\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta$

Solve the equation for exact solutions. -In an electric circuit, the electromotive force is defined as $V = \cos 2 \pi t$ , where $t$ is time in seconds. Solve the equation for $\mathrm { t }$ .

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. -A formula for the up-and-down motion of a weight on a spring is given by $\mathrm { S } = 2 \sin \left( \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \right) \mathrm { t }$ , where $\mathrm { k }$ is the spring constant, $\mathrm { m }$ is the mass, and $\mathrm { t }$ is the time. Solve the equation for $\mathrm { t }$ .

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { 1 } { 1 - \cos x } + \frac { 1 } { 1 + \cos x } = \csc ^ { 2 } x$

Use the fundamental identities to simplify the expression. - $\frac { \csc \theta \cot \theta } { \sec \theta }$

Use the fundamental identities to simplify the expression. - $\cot \theta \sec \theta \sin \theta$

Use a calculator to give the value to the nearest degree. - $\theta = \sin ^ { - 1 } ( 0.6561 )$

Use identities to write the expression as a single function of x or $\theta$ - $\tan ( x + \pi )$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sec \theta$ if $\tan \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant $I$ .

Verify that each equation is an identity. - $\csc s - \sin s = \cos s \cot s$

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\cos ^ { 2 } \frac { \theta } { 2 } = 1$

Perform the transformation. -Write $\csc x$ in terms of $\cot x$ .

Find the exact value of the real number y. - $\mathrm { y } = \arctan ( 1 )$

Solve the equation for exact solutions. - $\arctan x = \operatorname { arccot } \frac { 7 } { 16 }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\tan \theta$ if $\sin \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant II.

Decide whether the expression is or is not an identity. - $\sqrt { 1 - \cos ^ { 2 } x } = \sin x$

Write the product as a sum or difference of trigonometric functions. - $10 \sin 49 ^ { \circ } \cos 111 ^ { \circ }$