Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 15

Chapter 3, Problem 15

Find all solutions of each equation. cos x = ﹣1/2

Verified step by step guidance

Recognize that the equation is \( \cos x = -\frac{1}{2} \). Our goal is to find all angles \( x \) where the cosine value is \( -\frac{1}{2} \).

Recall the unit circle values for cosine. Cosine corresponds to the x-coordinate on the unit circle. The value \( -\frac{1}{2} \) occurs at specific standard angles in the second and third quadrants.

Identify the reference angle \( \theta \) where \( \cos \theta = \frac{1}{2} \). This reference angle is \( \theta = \frac{\pi}{3} \) (or 60 degrees).

Since cosine is negative in the second and third quadrants, the solutions for \( x \) are \( x = \pi - \frac{\pi}{3} \) and \( x = \pi + \frac{\pi}{3} \).

Write the general solution by adding the full period of cosine, which is \( 2\pi \), to each solution: \( x = \pi \pm \frac{\pi}{3} + 2k\pi \), where \( k \) is any integer.

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Key Concepts

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

Inverse Cosine Function (Arccos)

The inverse cosine function, denoted as arccos or cos⁻¹, is used to find the angle whose cosine value is a given number. Since cosine is not one-to-one over all real numbers, arccos returns values in the principal range [0, π]. This concept helps determine the initial solution(s) for the equation cos x = -1/2.

General Solutions of Trigonometric Equations

Trigonometric equations often have infinitely many solutions due to the periodic nature of trig functions. For cosine, solutions repeat every 2π. After finding the principal solution using arccos, the general solutions are given by x = ± arccos(value) + 2kπ, where k is any integer, capturing all possible angles.

Unit Circle and Cosine Values

The unit circle represents angles and their cosine values as the x-coordinate of points on the circle. Understanding where cosine equals -1/2 on the unit circle (in the second and third quadrants) helps visualize and identify the specific angle measures that satisfy the equation cos x = -1/2.

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Related Practice

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Use the given information to find the exact value of each of the following: sin 2θ

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Textbook Question

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Textbook Question

In Exercises 12–18, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 sin² x + cos x = 1