Chapter 7: Q. 74 (page 465)

In Problems 57– 80, solve each equation on the interval $0 \leq θ \leq 2 π$
$2 \cos^{2} θ - 7 \cos θ - 4 = 0$

Short Answer

Expert verified

The solution set of the given equation is $\{\frac{2 π}{3}, \frac{4 π}{3}\}$

Step by step solution

Step 1. Given Information

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

$2 \cos^{2} θ - 7 \cos θ - 4 = 0$

Step 2. The equation in its present form contains a cosine.

Factor the equation.

$2 \cos^{2} θ - (8 - 1) \cos θ - 4 = 0 2 \cos^{2} θ - 8 \cos θ + \cos θ - 4 = 0 2 \cos θ (\cos θ - 4) + 1 (\cos θ - 4) = 0 (2 \cos θ + 1) (\cos θ - 4) = 0$

Step 3. Use the Zero-Product Property.

$2 \cos θ + 1 = 0 or \cos θ - 4 = 0 2 \cos θ + 1 - 1 = 0 - 1 or \cos θ - 4 + 4 = 0 + 4 2 \cos θ = - 1 or \cos θ = 4 \frac{2}{2} \cos θ = - \frac{1}{2} or \cos θ = 4 \cos θ = - \frac{1}{2} or \cos θ = 4$

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

$θ = \frac{2 π}{3}, \frac{4 π}{3} or No solution$

The solution set is $\{\frac{2 π}{3}, \frac{4 π}{3}\}$

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In Problems 57– 80, solve each equation on the interval $0 \leq θ \leq 2 π$
$2 \cos^{2} θ - 7 \cos θ - 4 = 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The equation in its present form contains a cosine.

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-7cosθ-4=0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The equation in its present form contains a cosine.

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

Logic and Functions

Calculus

Decision Maths

Discrete Mathematics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 57– 80, solve each equation on the interval $0 \leq θ \leq 2 π$
$2 \cos^{2} θ - 7 \cos θ - 4 = 0$