Chapter 7

The terminal side of an angle in standard position passes through the given point. Sketch the angle, compute the distance \(r\) from the orgin to the point, write the six trigonometric functions of the angle, and find the angle. Work to three significant digits. $$(2.25,4.82)$$

What force, neglecting friction, must be exerted to drag a \(56.5-\mathrm{N}\) weight up a slope inclined \(12.6^{\circ}\) from the horizontal?

Given the magnitude of each vector and the angle \(\theta\) that it makes with the \(x\) axis, find the \(x\) and \(y\) components. $$\text { Magnitude }=4.93 \quad \theta=48.3^{\circ}$$

From a point on the ground 255 m from the base of a tower, the angle of elevation to the top of the tower is \(57.6^{\circ} .\) Find the height of the tower.

The terminal side of an angle in standard position passes through the given point. Sketch the angle, compute the distance \(r\) from the orgin to the point, write the six trigonometric functions of the angle, and find the angle. Work to three significant digits. $$(1.74,2.88)$$

Given the magnitude of each vector and the angle \(\theta\) that it makes with the \(x\) axis, find the \(x\) and \(y\) components. $$\text { Magnitude }=835 \quad \theta=25.8^{\circ}$$

A pilot \(4220 \mathrm{m}\) directly above the front of a straight train observes that the angle of depression of the end of the train is \(68.2^{\circ} .\) Find the length of the train.

The terminal side of an angle in standard position passes through the given point. Sketch the angle, compute the distance \(r\) from the orgin to the point, write the six trigonometric functions of the angle, and find the angle. Work to three significant digits. $$(3.72,5.49)$$

Given the magnitude of each vector and the angle \(\theta\) that it makes with the \(x\) axis, find the \(x\) and \(y\) components. $$\text { Magnitude }=1.884 \quad \theta=58.24^{\circ}$$

Find the sine, cosine, and tangent. Keep four decimal places. (a) \(49.3^{\circ}\) (b) \(38.9^{\circ}\) (c) \(18.3^{\circ}\) (d) \(2.07^{\circ}\) (e) \(85.3^{\circ}\) (1) \(28.7^{\circ}\) (g) \(73.7^{\circ}\) (h) \(43.9^{\circ}\) (i) \(3.345^{\circ}\) (j) \(58.49^{\circ}\) (k) \(78.37^{\circ}\) (1) \(22.05^{\circ}\) (m) \(83^{\circ} 43^{\prime}\) (n) \(78^{\circ} 27^{\prime}\) (o) \(33^{\circ} 47^{\prime}\) (p) \(63^{\circ} 29^{\prime}\)

From the top of a lighthouse \(156 \mathrm{ft}\) above the surface of the water, the angle of depression of a boat is observed to be \(28.7^{\circ} .\) Find the horizontal distance from the boat to the lighthouse.

The terminal side of an angle in standard position passes through the given point. Sketch the angle, compute the distance \(r\) from the orgin to the point, write the six trigonometric functions of the angle, and find the angle. Work to three significant digits. $$(7.93,8.27)$$

A truck weighing 7280 lb is on a bridge inclined \(4.80^{\circ}\) from the horizontal. Find the force of the truck normal (perpendicular) to the bridge.

Given the magnitude of each vector and the angle \(\theta\) that it makes with the \(x\) axis, find the \(x\) and \(y\) components. $$\text { Magnitude }=362 \quad \theta=13.8^{\circ}$$

Find the angle in decimal degrees whose trigonometric function is given. Keep three significant digits. $$\sin A=0.500$$

An observer in an airplane \(1520 \mathrm{ft}\) above the surface of the ocean observes that the angle of depression of a ship is \(28.8^{\circ} .\) Find the straight-line distance from the plane to the ship.

Sketch each right triangle and find all of the missing parts. Assume the triangles to be labeled as in Fig. \(7-10 .\) Work to three significant digits. $$b=7.74 \quad A=22.5^{\circ}$$

The terminal side of an angle in standard position passes through the given point. Sketch the angle, compute the distance \(r\) from the orgin to the point, write the six trigonometric functions of the angle, and find the angle. Work to three significant digits. $$(1.93,4.83)$$

A person has just enough strength to pull a \(1270-\mathrm{N}\) weight up a certain slope. Neglecting friction, find the angle at which the slope is inclined to the horizontal if the person is able to exert a pull of \(551 \mathrm{N}\).

Given the magnitude of each vector and the angle \(\theta\) that it makes with the \(x\) axis, find the \(x\) and \(y\) components. $$\text { Magnitude }=836 \quad \theta=45.2^{\circ}$$

Find the angle in decimal degrees whose trigonometric function is given. Keep three significant digits. $$\tan D=1.53$$

Sketch each right triangle and find all of the missing parts. Assume the triangles to be labeled as in Fig. \(7-10 .\) Work to three significant digits. $$a=284 \quad A=64.7^{\circ}$$

The terminal side of an angle in standard position passes through the given point. Sketch the angle, compute the distance \(r\) from the orgin to the point, write the six trigonometric functions of the angle, and find the angle. Work to three significant digits. $$(7.27,3.77)$$

Find the rectangular components of each vector. $$3.85<22.2^{\circ}$$

The angle of elevation of the top of a building from a point on the ground 275 ft from its base is \(51.3^{\circ} .\) Find the height of the building.

Find the angle in decimal degrees whose trigonometric function is given. Keep three significant digits. $$\sin G=0.528$$

Find the rectangular components of each vector. $$22.7 \angle 64.9^{\circ}$$

Find the angle in decimal degrees whose trigonometric function is given. Keep three significant digits. $$\cos K=0.770$$

The angle of elevation of the top of a building from a point on the ground 75.0 yd from its base is \(28.0^{\circ} .\) How high is the building?

Find the rectangular components of each vector. $$943 \angle 18.4^{\circ}$$

From the top of a hill 125 ft above a stream, the angles of depression of a point on the near shore and of a point on the opposite shore are \(42.3^{\circ}\) and \(40.6^{\circ} .\) Find the width of the stream between these two points.

Find the rectangular components of each vector. $$18.4 \angle 77.3^{\circ}$$

From the top of a tree 15.0 m high on the shore of a pond, the angle of depression of a point on the other shore is \(6.70^{\circ} .\) What is the width of the pond?

Find the angle in decimal degrees whose trigonometric function is given. Keep three significant digits. $$\cos E=0.847$$

Sketch each right triangle and find all of the missing parts. Assume the triangles to be labeled as in Fig. \(7-10 .\) Work to three significant digits. $$b=82.4 \quad A=31.4^{\circ}$$

Find the rectangular components of each vector. $$283<38.5^{\circ}$$

Evaluate the following, giving your answer in decimal degrees to three significant digits. $$\arcsin 0.635$$

A certain escalator travels at a rate of \(10.6 \mathrm{m} / \mathrm{min}\), and its angle of inclination is \(32.5^{\circ} .\) What is the vertical component of the velocity? How long will it take a passenger to travel 10.0 m vertically?

In the following problems, the magnitudes \(A\) and \(B\) of two perpendicular vectors are given. Find the resultant and the angle that it makes with \(B\) $$A=483 \quad B=382$$

Evaluate the following, giving your answer in decimal degrees to three significant digits. $$\arccos 0.862$$

An observer at a point \(P\) on a coast sights a ship in a direction \(\mathrm{N} 43^{\circ} 15^{\prime} \mathrm{E}\) The ship is at the same time directly east of a point \(Q, 15.6 \mathrm{km}\) due north of \(P\) Find the distance of the ship from point \(P\) and from \(Q\)

Sketch each right triangle and find all missing parts. Work to three significant digits and express the angles in decimal degrees. $$a=382 \quad b=274$$

A projectile is launched at an angle of \(55.6^{\circ}\) to the horizontal and follows a straight path with a speed of 7550 m/min. Find the vertical and horizontal components of this velocity.

In the following problems, the magnitudes \(A\) and \(B\) of two perpendicular vectors are given. Find the resultant and the angle that it makes with \(B\) $$A=2.85 \quad B=4.82$$

A ship sailing parallel to a straight coast is directly opposite one of two lights on the shore. The angle between the lines of sight from the ship to these lights is \(27^{\circ} 50^{\prime},\) and it is known that the lights are 355 m apart. Find the perpendicular distance of the ship from the shore.

Evaluate the following, giving your answer in decimal degrees to three significant digits. $$\tan ^{-1} 2.85$$

Sketch each right triangle and find all missing parts. Work to three significant digits and express the angles in decimal degrees. $$a=3.88 \quad c=5.37$$

In the following problems, the magnitudes \(A\) and \(B\) of two perpendicular vectors are given. Find the resultant and the angle that it makes with \(B\) $$A=7364 \quad B=4837$$

Evaluate the following, giving your answer in decimal degrees to three significant digits. $$\sin ^{-1} 0.175$$

Sketch each right triangle and find all missing parts. Work to three significant digits and express the angles in decimal degrees. $$b=3.97 \quad c=4.86$$