Chapter 6: Q.49 (page 395)

Find the exact value of each of the remaining trigonometric functions of $θ$
$\cos θ = \frac{- 1}{3}, \frac{π}{2} < θ < π$

Short Answer

Expert verified

The values are

$\sin θ = \frac{2 \sqrt{2}}{3}, c s c θ = \frac{3}{2 \sqrt{2}}, s e c θ = - 3, \tan θ = - 2 \sqrt{2}, c o t θ = - \frac{1}{2 \sqrt{2}}$

Step by step solution

.Given information

$\cos θ = \frac{- 1}{3}, \frac{π}{2} < θ < π$ is given.

We have to find the exact value of each of the remaining trigonometric functions

Step 2. Finding the value sinθ

$W e k n o w t h a t \sin^{2} θ + \cos^{2} θ = 1, h e r e \cos θ = \frac{- 1}{3} \sin θ = \pm \sqrt{1 - \cos^{2} θ}, h e r e θ l i e s o n \frac{π}{2} < θ < π s o \cos θ i s o n s e c o n d q u a d r a n t a n d a l l o t h e r e x c e p t \sin θ a n d c s c θ a r e w i l l b e n e g a t i v e . h e n c e \sin θ = \sqrt{1 - c o s^{2} θ} = \sqrt{1 - \frac{{(- 1)}^{2}}{3^{2}}} = \sqrt{1 - \frac{1}{9}} = \sqrt{\frac{8}{9}} = \frac{2 \sqrt{2}}{3}$

Step 3.Find the value of secθ and cscθ

$S i n c e \cos θ = \frac{- 1}{3}, t h e n s e c θ = \frac{1}{\cos θ} = \frac{1}{\frac{- 1}{3}} = - 3 c s c θ = \frac{1}{\sin θ} = \frac{1}{\frac{2 \sqrt{2}}{3}} = \frac{3}{2 \sqrt{2}}$

Step 4. find the values of tanθ and cotθ

$w e k n o w t h a t \tan θ = \frac{\sin θ}{\cos θ}, \sin θ = \frac{2 \sqrt{2}}{3} a n d \cos θ = \frac{- 1}{3} = \frac{\frac{2 \sqrt{2}}{3}}{\frac{- 1}{3}} = - 2 \sqrt{2} c o t θ = \frac{1}{\tan θ} = \frac{1}{- 2 \sqrt{2}}$

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.

Find the exact value of each of the remaining trigonometric functions of $θ$
$\cos θ = \frac{- 1}{3}, \frac{π}{2} < θ < π$

Short Answer

Step by step solution

.Given information

Step 2. Finding the value sinθ

Step 3.Find the value of secθ and cscθ

Step 4. find the values of tanθ and cotθ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

Find the exact value of each of the remaining trigonometric functions of θcosθ=-13,π2<θ<π

Short Answer

Step by step solution

.Given information

Step 2. Finding the value sinθ

Step 3.Find the value of secθ and cscθ

Step 4. find the values of tanθ and cotθ

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

Find the exact value of each of the remaining trigonometric functions of $θ$
$\cos θ = \frac{- 1}{3}, \frac{π}{2} < θ < π$