Chapter 7: Q. 60 (page 465)

In Problems 57– 80, solve each equation on the interval $0 \leq θ \leq 2 π$
$2 \cos^{2} θ + \cos θ - 1 = 0$

Short Answer

Expert verified

The solution set of the given equation is $\{π, \frac{π}{3}, \frac{5 π}{3}\}$ .

Step by step solution

Step 1. Given Information

In the given problems we have to solve each equation on the interval $0 \leq θ \leq 2 π$ .

$2 \cos^{2} θ + \cos θ - 1 = 0$

Step 2. This equation is a quadratic equation (in cosθ) that can be factored.

$2 \cos^{2} θ + \cos θ - 1 = 0 2 \cos^{2} θ + (2 - 1) \cos θ - 1 = 0 2 \cos^{2} θ + 2 \cos θ - \cos θ - 1 = 0 2 \cos θ (\cos θ + 1) - 1 (\cos θ + 1) = 0 (2 \cos θ - 1) (\cos θ + 1) = 0$

Step 3. Use the Zero-Product Property.

$2 \cos θ - 1 = 0 or \cos θ + 1 = 0 2 \cos θ - 1 + 1 = 0 + 1 or \cos θ + 1 - 1 = 0 - 1 2 \cos θ = 1 or \cos θ = - 1 \frac{2}{2} \cos θ = \frac{1}{2} or \cos θ = - 1 \cos θ = \frac{1}{2} or \cos θ = - 1$

Step 4. Solving each equation in the interval [0,2π), we obtain

$θ = \frac{π}{3}, \frac{5 π}{3} or θ = π$

So the solution set is $\{π, \frac{π}{3}, \frac{5 π}{3}\}$ .

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In Problems 57– 80, solve each equation on the interval $0 \leq θ \leq 2 π$
$2 \cos^{2} θ + \cos θ - 1 = 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. This equation is a quadratic equation (in cosθ) that can be factored.

Step 3. Use the Zero-Product Property.

Step 4. Solving each equation in the interval [0,2π), we obtain

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In Problems 57– 80, solve each equation on the interval 0≤θ≤2π2cos2θ+cosθ-1=0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. This equation is a quadratic equation (in cosθ) that can be factored.

Step 3. Use the Zero-Product Property.

Step 4. Solving each equation in the interval [0,2π), we obtain

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

Logic and Functions

Decision Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

In Problems 57– 80, solve each equation on the interval $0 \leq θ \leq 2 π$
$2 \cos^{2} θ + \cos θ - 1 = 0$