Exam 7: Applications of Trigonometric Functions

Solve the problem. -Island A is 150 miles from island B. A ship captain travels 250 miles from island A and then finds that he is off course and 160 miles from island B. What angle, in degrees, must he turn through to head straight for island B? Round the answer to two decimal places. (Hint: Be careful to properly identify which angle is the turning Angle.)

Find the area of the triangle. If necessary, round the answer to two decimal places. - $\alpha = 20 ^ { \circ } , \mathrm { b } = 13 , \mathrm { c } = 4$

Choose the one alternative that best completes the statement or answers the question. -A new homeowner has a triangular-shaped back yard. Two of the three sides measure $65 \mathrm { ft }$ and $80 \mathrm { ft }$ and form an included angle of $125 ^ { \circ }$ . The owner wants to approximate the area of the yard, so that he can determine the amount of fertilizer and grass seed to be purchased. Find the area of the yard rounded to the nearest square foot.

Solve the problem. -A ship sailing parallel to shore sights a lighthouse at an angle of from its direction of travel. After traveling 3 $12 ^ { \circ }$ miles farther, the angle is . At that time, how far is the ship from the lighthouse? $20 ^ { \circ }$

Solve the problem. -A photographer points a camera at a window in a nearby building forming an angle of 42° with the camera platform. If the camera is 52 m from the building, how high above the platform is the window, to the nearest Hundredth of a meter?

Solve the problem. -A ship at sea, the Admiral, spots two other ships, the Barstow and the Cauldrew and measures the angle between them at be 45°. They radio the Barstow and by comparing known landmarks, the distance between the The Admiral and the Barstow is found to be 323 meters. The Barstow reports an angle of 59° between the Admiral and the Cauldrew. To the nearest meter, what is the distance between the Barstow and the Cauldrew?

Solve the problem. -A surveyor is measuring the distance across a small lake. He has set up his transit on one side of the lake 90 feet from a piling that is directly across from a pier on the other side of the lake. From his transit, the angle between The piling and the pier is 55°. What is the distance between the piling and the pier to the nearest foot?

Choose the one alternative that best completes the statement or answers the question. -Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor. They each sight a landmark on the canyon floor on a line directly between them. The angles of depression from each hiker to the landmark meter are $37 ^ { \circ }$ and $21 ^ { \circ }$ . How far apart are the hikers? Round your answer to the nearest whole meter.

Write the word or phrase that best completes each statement or answers the question. Solve the problem. -A square wave is built up from sinusoidal curves of varying periods and amplitudes. Graph the following function, which can be used to approximate the square wave. $f ( x ) = \frac { 4 } { \pi } \left[ \sin ( \pi x ) + \frac { 1 } { 3 } \right.$ $0 \leq x \leq 4$ A better approximation to the square wave is given by $f ( x ) = \frac { 4 } { \pi } \left[ \sin ( \pi x ) + \frac { 1 } { 3 } \sin ( 3 \pi x ) + \frac { 1 } { 5 } \sin ( 5 \pi x ) \right] 0 \leq x \leq 4$ Graph this function and compare the result to the previous graph. Adding another term will improve the approximation even more. Write this new function with four terms.

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. - $\mathrm { a } = 14 , \quad \mathrm {~b} = 12 , \quad \beta = 25 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Solve the triangle. -

Choose the one alternative that best completes the statement or answers the question. Find the area of the triangle. If necessary, round the answer to two decimal places. -

Solve the problem. -A tree casts a shadow of 26 meters when the angle of elevation of the sun is 24°. Find the height of the tree to the nearest meter.

Choose the one alternative that best completes the statement or answers the question. -From the edge of a 1000 -foot cliff, the angles of depression to two cars in the valley below are $21 ^ { \circ }$ and $28 ^ { \circ }$ . How far apart are the cars? Round your answers to the nearest $0.1 \mathrm { ft }$ .

Write the word or phrase that best completes each statement or answers the question. -Penrose tiles are formed from a rhombus WXYZ with sides of length 1 and interior angles $72 ^ { \circ }$ and $108 ^ { \circ }$ . (Refer to the illustration.) $A$ point $O$ is chosen on the diagonal 1 unit from $Y$ . Line segments $O X$ and $O Z$ are drawn to the o vertices. The two resulting tiles are called a kite (figure $\mathrm { OXYZ }$ ) and a dart (figure $\mathrm { OXWZ }$ ). Find the area of the kit and the dart tile, correct to the nearest $0.01$ .

Choose the one alternative that best completes the statement or answers the question. -A flagpole is perpendicular to the horizontal but is on a slope that rises 10° from the horizontal. The pole casts a 43-foot shadow down the slope and angle of elevation of the sun measured from the slope is 36°. How tall is the Pole? Round your answer to the nearest 0.1 foot.

Choose the one alternative that best completes the statement or answers the question. Find the area of the triangle. If necessary, round the answer to two decimal places. -a = 14, b = 32, c = 26

Choose the one alternative that best completes the statement or answers the question. Solve the triangle. -

An object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up. - $\mathrm { a } = 10 ; \quad \mathrm { T } = 3 \pi$ seconds

Solve the triangle. - $\beta = 10 ^ { \circ } , \quad \gamma = 100 ^ { \circ } , \quad \mathrm { a } = 4$