Chapter 7: Q. 59 (page 504)

In Problems $58 - 63$ , find the exact value of each expression.
$\sin (\cos^{- 1} \frac{5}{13} - \cos^{- 1} \frac{4}{5})$ .

Short Answer

Expert verified

The exact value of the expression $\sin (\cos^{- 1} \frac{5}{13} - \cos^{- 1} \frac{4}{5})$ is, $\frac{33}{65}$ .

Step by step solution

Step 1 Given expression is,

$\sin (\cos^{- 1} \frac{5}{13} - \cos^{- 1} \frac{4}{5})$ .

The formulae used in this problem are,

$\sin^{2} α + \cos^{2} α = 1 \sin (α - β) = \sin α \cos β - \cos α \sin β$

Let $α = \cos^{- 1} \frac{5}{13}$

$\cos α = \frac{5}{13}$

Step 2 Find the value of sin α.

$\sin α = \sqrt{1 - \cos^{2} α} = \sqrt{1 - {(\frac{5}{13})}^{2}} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} \sin α = \frac{12}{13}$

Step 3 Next find the value of cos β.

$\cos β = \sqrt{1 - \sin^{2} α} = \sqrt{1 - {(\frac{4}{5})}^{2}} = \sqrt{1 - \frac{16}{25}} \cos β = \frac{3}{5}$

Step 4 Substitute the known values in the cos α-β formula.

$\cos (α - β) = (\frac{12}{13}) (\frac{4}{5}) - (\frac{5}{13}) (\frac{3}{5}) = \frac{48}{65} - \frac{15}{65} = \frac{33}{65} \sin (\cos^{- 1} \frac{5}{13} - \cos^{- 1} \frac{4}{5}) = \frac{33}{65}$

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.

In Problems $58 - 63$ , find the exact value of each expression.
$\sin (\cos^{- 1} \frac{5}{13} - \cos^{- 1} \frac{4}{5})$ .

Short Answer

Step by step solution

Step 1 Given expression is,

Step 2 Find the value of sin α.

Step 3 Next find the value of cos β.

Step 4 Substitute the known values in the cos α-β formula.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Pure Maths

Mechanics Maths

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 58-63, find the exact value of each expression.sincos-1513-cos-145.

Short Answer

Step by step solution

Step 1 Given expression is,

Step 2 Find the value of sin α.

Step 3 Next find the value of cos β.

Step 4 Substitute the known values in the cos α-β formula.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Pure Maths

Mechanics Maths

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems $58 - 63$ , find the exact value of each expression.
$\sin (\cos^{- 1} \frac{5}{13} - \cos^{- 1} \frac{4}{5})$ .