Exam 5: Trigonometric Functions

Solve the problem. -Suppose that the average monthly low temperatures for a small town are shown in the table. \begin{tabular} { l | l l l l l l l l l l l l } Month & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline Temperature $\left( { } ^ { \circ } \mathrm { F } \right)$ & 19 & 27 & 38 & 45 & 57 & 62 & 65 & 58 & 51 & 41 & 33 & 25 \end{tabular} Model this data using $f ( x ) = a \sin ( b ( x - c ) ) + d$ . Use the sine regression feature to do this. Approximate all values to one decimal place.

Let θ be an angle in standard position. Name the quadrant in which the angle θ lies. - $\cos \theta = \frac { 2 } { 5 } , \quad \tan \theta < 0$ Find $\sin \theta$ .

Use the given triangles to evaluate the expression. Rationalize all denominators. - $\cos \frac { \pi } { 6 }$

Use the Pythagorean Theorem to find the length of the missing side.Then find the indicated trigonometric function of the given angle. Give an exact answer with a rational denominator. -Find $\tan \theta$ .

Solve the right triangle shown in the figure. Round lengths to one decimal place and express angles to the nearest tenth of a degree. -From a boat on the river below a dam, the angle of elevation to the top of the dam is 27°8'. If the dam is 1982 feet above the level of the river, how far is the boat from the base of the dam (to the nearest foot)?

Solve the problem. -The equation $\theta _ { 2 } = \tan ^ { - 1 } ( \omega \mathrm { C } / \mathrm { G } )$ gives the phase angle of impedance in the parallel portion of a distributed constant circuit. Find $\theta _ { 2 }$ if $\omega = 310$ radians per second, $\mathrm { C } = 0.04 \mu \mathrm { F }$ per kilometer, and $\mathrm { G } = 1.22 \mu$ siemens per kilometer.

The given angle is in standard position. Determine the quadrant in which the angle lies. - $18 ^ { \circ }$

Find the exact value of the expression. - $\tan ^ { - 1 } \sqrt { 3 }$

Use periodic properties of the trigonometric functions to find the exact value of the expression. - $\tan 9 \pi$

Find the exact value of the expression, if possible. Do not use a calculator. - $\tan \left( \tan ^ { - 1 } ( - 5.8 ) \right)$

Use reference angles to find the exact value of the expression. Do not use a calculator. - $\csc \frac { 3 \pi } { 2 }$

Determine the amplitude or period as requested. -Period of $y = - 2 \sin 6 \pi x$

Find a cofunction with the same value as the given expression. - $\sin 20 ^ { \circ }$

A point on the terminal side of angle θ is given. Find the exact value of the indicated trigonometric function of θ. - $( - 10,24 )$ Find $\sin \theta$ .

Graph the function. - $y=2 \cos (3 x-\pi)$

Use even and odd properties of the trigonometric functions to find the exact value of the expression. - $\cot \left( - \frac { \pi } { 6 } \right)$

Find the exact value of the expression. - $\sin ^ { - 1 } \frac { \sqrt { 3 } } { 2 }$

Use even and odd properties of the trigonometric functions to find the exact value of the expression. - $\sin \left( - \frac { \pi } { 4 } \right)$

An object is attached to a coiled spring. The object is pulled down (negative direction from the rest position) and then released. Write an equation for the distance of the object from its rest position after t seconds. -An object in simple harmonic motion has a frequency of $\frac { 5 } { 2 }$ oscillations per second and an amplitude of 3 feet. Write an equation in the form $\mathrm { d } = \mathrm { a } \sin \omega t$ for the object's simple harmonic motion.

Using a calculator, solve the following problems. Round your answers to the nearest tenth. -A ship is 9 miles west and 11 miles south of a harbor. What bearing should the captain set to sail directly to harbor?