Chapter 7: Q. 96 (page 487)

solve each equation on the interval $0 \leq θ < 2 π$
$c o t θ + c s c θ = - \sqrt{3}$

Short Answer

Expert verified

The solution of $c o t θ + c s c θ = - \sqrt{3}$ in interval $0 \leq θ < 2 π$ is $θ = \frac{5 π}{3}$

Step by step solution

Step 1. Given data

The given equation is

$c o t θ + c s c θ = - \sqrt{3}$

The given interval is

$0 \leq θ < 2 π$

Step 2. Formation of the proper equation

Rearrange the equation

$c o t θ + c s c θ = - \sqrt{3} \frac{\cos θ}{\sin θ} + \frac{1}{\sin θ} = - \sqrt{3} \cos θ + 1 = - \sqrt{3} \sin θ \cos θ + \sqrt{3} \sin θ = - 1 \frac{1}{2} \cos θ + \frac{\sqrt{3}}{2} \sin θ = - \frac{1}{2}$

Step 3. Use of sum formula of sin function

Rearrange the equation and use the sum formula of the sine function

$\frac{1}{2} \cos θ + \frac{\sqrt{3}}{2} \sin θ = - \frac{1}{2} S i n \frac{π}{6} \cdot \cos θ + \cos \frac{π}{6} \cdot \sin θ = S i n \frac{7 π}{6} \sin (\frac{π}{6} + θ) = S i n \frac{7 π}{6} \frac{π}{6} + θ = \frac{7 π}{6} θ = π$

But the equation is not true for $θ = π$ so it is not the solution

Step 4. Use of sum formula of sin function

Rearrange the equation and use the sum formula of the sine function

$\frac{1}{2} \cos θ + \frac{\sqrt{3}}{2} \sin θ = - \frac{1}{2} S i n \frac{π}{6} \cdot \cos θ + \cos \frac{π}{6} \cdot \sin θ = S i n \frac{11 π}{6} \sin (\frac{π}{6} + θ) = S i n \frac{11 π}{6} \frac{π}{6} + θ = \frac{11 π}{6} θ = \frac{10 π}{6} θ = \frac{5 π}{3}$

The equation is true for $θ = \frac{5 π}{3}$

so solution is $θ = \frac{5 π}{3}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

solve each equation on the interval $0 \leq θ < 2 π$
$c o t θ + c s c θ = - \sqrt{3}$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Formation of the proper equation

Step 3. Use of sum formula of sin function

Step 4. Use of sum formula of sin function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

solve each equation on the interval 0≤θ<2πcotθ+cscθ=-3

Short Answer

Step by step solution

Step 1. Given data

Step 2. Formation of the proper equation

Step 3. Use of sum formula of sin function

Step 4. Use of sum formula of sin function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

solve each equation on the interval $0 \leq θ < 2 π$
$c o t θ + c s c θ = - \sqrt{3}$