Chapter 7: Q. 80 (page 465)

In Problems 57– 80, solve each equation on the interval $0 \leq θ \leq 2 π$
$\sec θ = \tan θ + \cot θ$

Short Answer

Expert verified

The solution set of the given equation is $\{\frac{π}{2}\}$ .

Step by step solution

Step 1. Given Information

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

localid="1647016175573" $\sec θ = \tan θ + \cot θ$

Step 2. The equation in its present form contains a secant, cotangent and tangent.

We know that $\sec θ = \frac{1}{\cos θ}, \tan θ = \frac{\sin θ}{\cos θ}, \cot θ = \frac{\cos θ}{\sin θ}$

Step 3. Putting the value in the given equation.

$\sec θ = \tan θ + \cot θ \frac{1}{\cos θ} = \frac{\sin θ}{\cos θ} + \frac{\cos θ}{\sin θ}$

The LCD is role="math" localid="1647016289932" $\cos θ \cdot \sin θ$ .

Step 4. Multiply with the LCD on both side.

$\frac{1}{\cos θ} \cdot \cos θ \cdot \sin θ = \frac{\sin θ}{\cos θ} \cdot \cos θ \cdot \sin θ + \frac{\cos θ}{\sin θ} \cdot \cos θ \cdot \sin θ \sin θ = \sin θ \cdot \sin θ + \cos θ \cdot \cos θ \sin θ = \sin^{2} θ + \cos^{2} θ$

Step 5. We know that sin2θ+cos2θ=1

So $\sin θ = 1$

Solving each equation in the interval $[0, 2 π]$ , we obtain

$θ = \frac{π}{2}$

So the solution set is $\{\frac{π}{2}\}$ .

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In Problems 57– 80, solve each equation on the interval $0 \leq θ \leq 2 π$
$\sec θ = \tan θ + \cot θ$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The equation in its present form contains a secant, cotangent and tangent.

Step 3. Putting the value in the given equation.

Step 4. Multiply with the LCD on both side.

Step 5. We know that sin2θ+cos2θ=1

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In Problems 57– 80, solve each equation on the interval 0≤θ≤2πsecθ=tanθ+cotθ

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The equation in its present form contains a secant, cotangent and tangent.

Step 3. Putting the value in the given equation.

Step 4. Multiply with the LCD on both side.

Step 5. We know that sin2θ+cos2θ=1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Probability and Statistics

Decision Maths

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 57– 80, solve each equation on the interval $0 \leq θ \leq 2 π$
$\sec θ = \tan θ + \cot θ$