Q. 36 find the exact value of each of ... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 36 (page 486)

find the exact value of each of the following under the given conditions : $\tan 伪 = \frac{5}{12}, 蟺 < 伪 < \frac{3 蟺}{2}; \sin 尾 = - \frac{1}{2}, 蟺 < 尾 < \frac{3 蟺}{2}$
$a) \sin (伪 + 尾) b) \cos (伪 + 尾) c) \sin (伪 - 尾) d) \tan (伪 - 尾)$

Short Answer

Expert verified

The values are

$a) \sin (伪 + 尾) = \frac{12 + 5 \sqrt{3}}{26} b) \cos (伪 + 尾) = \frac{12 \sqrt{3} - 5}{26} c) \sin (伪 - 尾) = \frac{5 \sqrt{3} - 12}{26} d) \tan (伪 - 尾) = \frac{5 - 12 \sqrt{3}}{12 + 5 \sqrt{3}}$

Step by step solution

Given information

We are given $\tan 伪 = \frac{5}{12}, 蟺 < 伪 < \frac{3 蟺}{2}; \sin 尾 = - \frac{1}{2}, 蟺 < 尾 < \frac{3 蟺}{2}$

We find cosα, sinα

We have,

$\cos 伪 = \sqrt{\frac{1}{\tan^{2} 伪 + 1}} \cos 伪 = \sqrt{\frac{1}{{(\frac{5}{12})}^{2} + 1}} \cos 伪 = - \frac{12}{13} w h e r e 伪 i s i n t h i r d q u a d r a n t$

Similarly

$\sin 伪 = \sqrt{1 - \cos^{2} 伪} \sin 伪 = \sqrt{1 - {(- \frac{12}{13})}^{2}} \sin 伪 = - \frac{5}{13}$

Now we find cosβ

We have,

$\cos 尾 = \sqrt{1 - \sin^{2} 尾} \cos 尾 = \sqrt{1 - {(- \frac{1}{2})}^{2}} \cos 尾 = - \frac{\sqrt{3}}{2}$

Find sin(α+β)

We have,

$\sin (伪 + 尾) = \sin 伪 \cos 尾 + \cos 伪 \sin 尾 \sin (伪 + 尾) = (- \frac{5}{13}) (- \frac{\sqrt{3}}{2}) + (- \frac{12}{13}) (- \frac{1}{2}) \sin (伪 + 尾) = \frac{5 \sqrt{3}}{26} + \frac{12}{26} \sin (伪 + 尾) = \frac{12 + 5 \sqrt{3}}{26}$

Find cos(α+β)

We have,

$co s (伪 + 尾) = \cos 伪 \cos 尾 - \sin 伪 \sin 尾 co s (伪 + 尾) = (- \frac{12}{13}) (- \frac{\sqrt{3}}{2}) - (- \frac{5}{13}) (- \frac{1}{2}) co s (伪 + 尾) = \frac{12 \sqrt{3}}{26} - \frac{5}{26} co s (伪 + 尾) = \frac{12 \sqrt{3} - 5}{26}$

Find sin(α-β)

We have,

$\sin (伪 - 尾) = \sin 伪 \cos 尾 - \cos 伪 \sin 尾 \sin (伪 - 尾) = (- \frac{5}{13}) (- \frac{\sqrt{3}}{2}) - (- \frac{12}{13}) (- \frac{1}{2}) \sin (伪 - 尾) = \frac{5 \sqrt{3}}{26} - \frac{12}{26} \sin (伪 - 尾) = \frac{5 \sqrt{3} - 12}{26}$

find tan(α-β)

We have,

$\tan 尾 = \frac{\sin 尾}{\cos 尾} \tan 尾 = \frac{- \frac{1}{2}}{- \frac{\sqrt{3}}{2}} \tan 尾 = \frac{1}{\sqrt{3}}$

Now

$\tan (伪 - 尾) = \frac{\tan 伪 - \tan 尾}{1 + \tan 伪 \tan 尾} \tan (伪 - 尾) = \frac{\frac{5}{12} - \sqrt{3}}{1 + (\frac{5}{12}) (\sqrt{3})} \tan (伪 - 尾) = \frac{5 - 12 \sqrt{3}}{12 + 5 \sqrt{3}}$

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91影视

find the exact value of each of the following under the given conditions : $\tan 伪 = \frac{5}{12}, 蟺 < 伪 < \frac{3 蟺}{2}; \sin 尾 = - \frac{1}{2}, 蟺 < 尾 < \frac{3 蟺}{2}$
$a) \sin (伪 + 尾) b) \cos (伪 + 尾) c) \sin (伪 - 尾) d) \tan (伪 - 尾)$

Short Answer

Step by step solution

Given information

We find cosα, sinα

Now we find cosβ

Find sin(α+β)

Find cos(α+β)

Find sin(α-β)

find tan(α-β)

One App. One Place for Learning.

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91影视

find the exact value of each of the following under the given conditions : tan伪=512,蟺<伪<3蟺2;sin尾=-12,蟺<尾<3蟺2a)sin(伪+尾)b)cos(伪+尾)c)sin(伪-尾)d)tan(伪-尾)

Short Answer

Step by step solution

Given information

We find cosα, sinα

Now we find cosβ

Find sin(α+β)

Find cos(α+β)

Find sin(α-β)

find tan(α-β)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

find the exact value of each of the following under the given conditions : $\tan 伪 = \frac{5}{12}, 蟺 < 伪 < \frac{3 蟺}{2}; \sin 尾 = - \frac{1}{2}, 蟺 < 尾 < \frac{3 蟺}{2}$
$a) \sin (伪 + 尾) b) \cos (伪 + 尾) c) \sin (伪 - 尾) d) \tan (伪 - 尾)$