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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