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Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblema 2.3.66
Capitolo 3, Problema 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

Guida verificata passo dopo passo
1
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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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.
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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°).
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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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Domanda del libro di testo

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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Domanda del libro di testo

(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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Domanda del libro di testo

Solve each problem. See Examples 1 and 2. Distance Traveled by a Ship A ship travels 55 km on a bearing of 27° and then travels on a bearing of 117° for 140 km. Find the distance from the starting point to the ending point.

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Domanda del libro di testo

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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Domanda del libro di testo

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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Domanda del libro di testo

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.

sin θ = 0.84802194

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