To determine To find: The exact value of expression under the given condition: a. sin ( α + β ) Answer Solution: The exact value of expressions under the given condition. a. sin ( ά + β ) = 6 5 25 − 4 5 5 Explanation Given: sin ά = 3 5 , 0 < ά < π 2 , I quadrant, cos ά = 1 − ( 3 5 ) 2 = + 4 5 cos β = 2 5 5 , − π 2 < β < 0 , IV quadrant, sin β = 1 − ( 2 5 5 ) 2 = − 1 5 Formula used: Sum formula of sine , sin ( α + β ) = sin α cos β + cos α sin β Calculation: a. sin ( ά + β ) = sin α cos β + cos α sin β sin ( ά + β ) = 3 5 2 5 5 + 4 5 ( − 1 5 ) sin ( ά + β ) = 6 5 25 − 4 5 5 To determine To find: The exact value of expression under the given condition: b. cos ( α + β ) Answer Solution: The exact value of expressions under the given condition. b. cos ( ά + β ) = 8 5 25 + 3 5 5 Explanation Given: sin ά = 3 5 , 0 < ά < π 2 , I quadrant, cos ά = 1 − ( 3 5 ) 2 = + 4 5 cos β = 2 5 5 , − π 2 < β < 0 , IV quadrant, sin β = 1 − ( 2 5 5 ) 2 = − 1 5 Formula used: Sum formula of cosine , cos ( α + β ) = cos α cos β − sin α sin β Calculation: b. cos ( ά + β ) = cos α cos β − sin α sin β cos ( ά + β ) = 4 5 2 5 5 − 3 5 ( − 1 5 ) cos ( ά + β ) = 8 5 25 + 3 5 5 To determine To find: The exact value of expression under the given condition: c. sin ( α − β ) Answer Solution: The exact value of expressions under the given condition. c. sin ( ά − β ) = 6 5 25 + 4 5 5 Explanation Given: sin ά = 3 5 , 0 < ά < π 2 , I quadrant, cos ά = 1 − ( 3 5 ) 2 = + 4 5 cos β = 2 5 5 , − π 2 < β < 0 , IV quadrant, sin β = 1 − ( 2 5 5 ) 2 = − 1 5 Formula used: Difference formula of sine , sin ( α − β ) = sin α cos β − cos α sin β Calculation: c. sin ( ά − β ) = sin α cos β − cos α sin β sin ( ά − β ) = 3 5 2 5 5 − 4 5 ( − 1 5 ) sin ( ά − β ) = 6 5 25 + 4 5 5 By quotient identity. To determine To find: The exact value of expression under the given condition: d. tan ( α − β ) Answer Solution: The exact value of expressions under the given condition. d. tan ( ά − β ) = 6 5 25 + 4 5 5 8 5 25 − 3 5 5 Explanation Given: sin ά = 3 5 , 0 < ά < π 2 , I quadrant, cos ά = 1 − ( 3 5 ) 2 = + 4 5 cos β = 2 5 5 , − π 2 < β < 0 , IV quadrant, sin β = 1 − ( 2 5 5 ) 2 = − 1 5 Formula used: Difference formula of tan , tan ( ά − β ) = sin ( ά − β ) cos ( ά − β ) Calculation: d. tan ( ά − β ) = sin ( ά − β ) cos ( ά − β ) = 6 5 25 + 4 5 5 8 5 25 − 3 5 5

9th Edition

ISBN: 9780321716835

Author: Michael Sullivan

Publisher: Addison Wesley

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Textbook Question

Chapter 7.5, Problem 33AYU

In Problems 35-40, find the exact value of each of the following under the given conditions:

(a) $\sin (α + β)$

(b) $\cos (α + β)$

(d) $\tan (α - β)$

$\sin α = \frac{3}{5}, 0 < α < \frac{π}{2}; \cos β = \frac{2 \sqrt{5}}{5}, - \frac{π}{2} < β < 0$

Expert Solution

To determine

To find: The exact value of expression under the given condition:

a. $\sin (α + β)$

Answer to Problem 33AYU

Solution:

The exact value of expressions under the given condition.

a. $\sin (ά + β) = \frac{6 \sqrt{5}}{25} - \frac{4}{5 \sqrt{5}}$

Explanation of Solution

Given:

$\sin ά = \frac{3}{5}$ , $0 < ά < \frac{π}{2}$ , I quadrant, $\cos ά = \sqrt{1 - {(\frac{3}{5})}^{2}} = + \frac{4}{5}$

$\cos β = \frac{2 \sqrt{5}}{5}$ , $- \frac{π}{2} < β < 0$ , IV quadrant, $sin β = \sqrt{1 - {(\frac{2 \sqrt{5}}{5})}^{2}} = - \frac{1}{\sqrt{5}}$

Formula used:

Sum formula of $sine$ ,

$\sin (α + β) = \sin α \cos β + \cos α \sin β$

Calculation:

a. $\sin (ά + β) = \sin α \cos β + \cos α \sin β$

$\sin (ά + β) = \frac{3}{5} \frac{2 \sqrt{5}}{5} + \frac{4}{5} (- \frac{1}{\sqrt{5}})$

$\sin (ά + β) = \frac{6 \sqrt{5}}{25} - \frac{4}{5 \sqrt{5}}$

Expert Solution

To determine

To find: The exact value of expression under the given condition:

b. $\cos (α + β)$

Answer to Problem 33AYU

Solution:

The exact value of expressions under the given condition.

b. $\cos (ά + β) = \frac{8 \sqrt{5}}{25} + \frac{3}{5 \sqrt{5}}$

Explanation of Solution

Given:

$\sin ά = \frac{3}{5}$ , $0 < ά < \frac{π}{2}$ , I quadrant, $\cos ά = \sqrt{1 - {(\frac{3}{5})}^{2}} = + \frac{4}{5}$

$\cos β = \frac{2 \sqrt{5}}{5}$ , $- \frac{π}{2} < β < 0$ , IV quadrant, $sin β = \sqrt{1 - {(\frac{2 \sqrt{5}}{5})}^{2}} = - \frac{1}{\sqrt{5}}$

Formula used:

Sum formula of $cosine$ ,

$\cos (α + β) = \cos α \cos β - \sin α \sin β$

Calculation:

b. $\cos (ά + β) = \cos α \cos β - \sin α \sin β$

$\cos (ά + β) = \frac{4}{5} \frac{2 \sqrt{5}}{5} - \frac{3}{5} (- \frac{1}{\sqrt{5}})$

$\cos (ά + β) = \frac{8 \sqrt{5}}{25} + \frac{3}{5 \sqrt{5}}$

Expert Solution

To determine

To find: The exact value of expression under the given condition:

c. $\sin (α - β)$

Answer to Problem 33AYU

Solution:

The exact value of expressions under the given condition.

c. $\sin (ά - β) = \frac{6 \sqrt{5}}{25} + \frac{4}{5 \sqrt{5}}$

Explanation of Solution

Given:

$\sin ά = \frac{3}{5}$ , $0 < ά < \frac{π}{2}$ , I quadrant, $\cos ά = \sqrt{1 - {(\frac{3}{5})}^{2}} = + \frac{4}{5}$

$\cos β = \frac{2 \sqrt{5}}{5}$ , $- \frac{π}{2} < β < 0$ , IV quadrant, $sin β = \sqrt{1 - {(\frac{2 \sqrt{5}}{5})}^{2}} = - \frac{1}{\sqrt{5}}$

Formula used:

Difference formula of $sine$ ,

$\sin (α - β) = \sin α \cos β - \cos α \sin β$

Calculation:

c. $\sin (ά - β) = \sin α \cos β - \cos α \sin β$

$\sin (ά - β) = \frac{3}{5} \frac{2 \sqrt{5}}{5} - \frac{4}{5} (- \frac{1}{\sqrt{5}})$

$\sin (ά - β) = \frac{6 \sqrt{5}}{25} + \frac{4}{5 \sqrt{5}}$

By quotient identity.

Expert Solution

To determine

To find: The exact value of expression under the given condition:

d. $\tan (α - β)$

Answer to Problem 33AYU

Solution:

The exact value of expressions under the given condition.

d. $\tan (ά - β) = \frac{\frac{6 \sqrt{5}}{25} + \frac{4}{5 \sqrt{5}}}{\frac{8 \sqrt{5}}{25} - \frac{3}{5 \sqrt{5}}}$

Explanation of Solution

Given:

$\sin ά = \frac{3}{5}$ , $0 < ά < \frac{π}{2}$ , I quadrant, $\cos ά = \sqrt{1 - {(\frac{3}{5})}^{2}} = + \frac{4}{5}$

$\cos β = \frac{2 \sqrt{5}}{5}$ , $- \frac{π}{2} < β < 0$ , IV quadrant, $sin β = \sqrt{1 - {(\frac{2 \sqrt{5}}{5})}^{2}} = - \frac{1}{\sqrt{5}}$

Formula used:

Difference formula of $\tan$ ,

$\tan (ά - β) = \frac{\sin (ά - β)}{\cos (ά - β)}$

Calculation:

d. $\tan (ά - β) = \frac{\sin (ά - β)}{\cos (ά - β)} = \frac{\frac{6 \sqrt{5}}{25} + \frac{4}{5 \sqrt{5}}}{\frac{8 \sqrt{5}}{25} - \frac{3}{5 \sqrt{5}}}$

Chapter 7.5, Problem 32AYU

Chapter 7.5, Problem 34AYU

Chapter 7 Solutions

Precalculus

Ch. 7.1 - Prob. 1AYU Ch. 7.1 - Prob. 2AYU Ch. 7.1 - Prob. 3AYU Ch. 7.1 - Prob. 4AYU Ch. 7.1 - Prob. 5AYU Ch. 7.1 - Prob. 6AYU Ch. 7.1 - y= sin 1 x means _____, where 1x1 and 2 y 2 .Ch. 7.1 - Prob. 8AYU Ch. 7.1 - Prob. 9AYU Ch. 7.1 - Prob. 10AYU

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