Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.3.66

Chapter 3, Problem 2.3.66

Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree. cos θ = 0.10452846

Verified step by step guidance

Recall that the cosine function is positive in the first and fourth quadrants within the interval \([0^\circ, 360^\circ)\).

Use the inverse cosine function to find the principal angle \(\theta_1\) by calculating \(\theta_1 = \cos^{-1}(0.10452846)\).

Calculate the second angle \(\theta_2\) by using the fact that cosine is positive in the fourth quadrant, so \(\theta_2 = 360^\circ - \theta_1\).

Round both \(\theta_1\) and \(\theta_2\) to the nearest degree as required by the problem.

Verify that both angles lie within the interval \([0^\circ, 360^\circ)\) and satisfy the original equation \(\cos \theta = 0.10452846\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosine Function and Its Values

The cosine function relates an angle in a right triangle to the ratio of the adjacent side over the hypotenuse. On the unit circle, cosine corresponds to the x-coordinate of a point at a given angle. Understanding how to interpret cosine values helps in finding angles that satisfy a given cosine equation.

Inverse Cosine Function (Arccos)

The inverse cosine function, arccos, is used to find the principal angle whose cosine is a given value. Since cosine is positive in the first and fourth quadrants, arccos returns an angle in [0°, 180°], and additional steps are needed to find all solutions within [0°, 360°).

General Solutions for Trigonometric Equations in [0°, 360°)

Trigonometric equations often have multiple solutions within one full rotation. For cosine, if θ is a solution, then 360° - θ is also a solution because cosine is symmetric about the x-axis. Identifying both solutions ensures all valid angles in the interval are found.

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Related Practice

Textbook Question

Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.

cos θ = 0.85536428

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Textbook Question

(Modeling) Fish's View of the World The figure shows a fish's view of the world above the surface of the water. (Data from Walker, J., 'The Amateur Scientist,' Scientific American.) Suppose that a light ray comes from the horizon, enters the water, and strikes the fish's eye. Assume that this ray gives a value of 90° for angle θ₁ in the formula for Snell's law. (In a practical situation, this angle would probably be a little less than 90°.) The speed of light in water is about 2.254 x 10⁸ m per sec. Find angle θ₂ to the nearest tenth.

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Textbook Question

Solve each problem. See Examples 1 and 2. Distance between Two Cities The bearing from Atlanta to Macon is S 27° E, and the bearing from Macon to Augusta is N 63° E. An automobile traveling at 62 mph needs 1¼ hr to go from Atlanta to Macon and 1¾ hr to go from Macon to Augusta. Find the distance from Atlanta to Augusta.

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Textbook Question

CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.

Column I: 1.

cot⁻¹ 30

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858

F. 1.962610506

G. 14.47751219°

H. 1.015426612

I. 1.051462224

J. 0.9925461516

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Textbook Question

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (0, -2)

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Textbook Question

Solve each problem. See Examples 3 and 4. Height of an Antenna A scanner antenna is on top of the center of a house. The angle of elevation from a point 28.0 m from the center of the house to the top of the antenna is 27°10', and the angle of elevation to the bottom of the antenna is 18°10'. Find the height of the antenna.

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