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 17

Chapter 3, Problem 17

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. cos θ = -½

Verified step by step guidance

Recall that the cosine function corresponds to the x-coordinate on the unit circle for an angle \( \theta \). We are given \( \cos \theta = -\frac{1}{2} \), and we need to find all \( \theta \) in the interval \( [0^\circ, 360^\circ) \) that satisfy this.

Identify the reference angle where \( \cos \theta = \frac{1}{2} \). Since cosine is positive in the first quadrant, find the acute angle \( \alpha \) such that \( \cos \alpha = \frac{1}{2} \). This reference angle \( \alpha \) is \( 60^\circ \) because \( \cos 60^\circ = \frac{1}{2} \).

Since the cosine value is negative, \( \theta \) must lie in the quadrants where cosine is negative. Cosine is negative in the second and third quadrants.

Use the reference angle to find the solutions in the second and third quadrants: - In the second quadrant, \( \theta = 180^\circ - \alpha \). - In the third quadrant, \( \theta = 180^\circ + \alpha \).

Write the general solutions explicitly using the reference angle \( 60^\circ \): - \( \theta = 180^\circ - 60^\circ \) - \( \theta = 180^\circ + 60^\circ \) These are the values of \( \theta \) in the interval \( [0^\circ, 360^\circ) \) where \( \cos \theta = -\frac{1}{2} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Unit Circle and Angle Measurement

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles in trigonometry are often measured in degrees or radians on this circle, where the cosine of an angle corresponds to the x-coordinate of the point on the circle. Understanding the unit circle helps identify where cosine values are positive or negative.

Cosine Function and Its Values

The cosine function relates an angle to the ratio of the adjacent side over the hypotenuse in a right triangle, or equivalently, the x-coordinate on the unit circle. Knowing that cos θ = -½ means finding angles where the x-coordinate is -0.5, which occurs in specific quadrants where cosine is negative.

Solving Trigonometric Equations in a Given Interval

To solve cos θ = -½ within [0°, 360°), identify all angles in this range where the cosine equals -½. Since cosine is negative in the second and third quadrants, use reference angles and symmetry properties to find all solutions within the specified interval.

Recommended video:

4:34

How to Solve Linear Trigonometric Equations

Related Practice

Textbook Question

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. b = 8, c = 11

views

Textbook Question

Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. 300°

views

Textbook Question

Find exact values of the six trigonometric functions for each angle. Do not use a calculator. Rationalize denominators when applicable. 120°

views

Textbook Question

Solve each right triangle. When two sides are given, give angles in degrees and minutes.

Textbook Question

Find all values of θ, if θ is in the interval [0°, 360°) and has the given function value. sec θ = -2√3 3

views

Textbook Question

views