Q. 60 In Problems 58-63, find the exac... [FREE SOLUTION]

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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 60 (page 504)

In Problems $58 - 63$ , find the exact value of each expression.
$\tan (\sin^{- 1} (\frac{- 1}{2}) - \tan^{- 1} \frac{3}{4})$ .

Short Answer

Expert verified

The exact value of the expression $\tan (\sin^{- 1} (\frac{- 1}{2}) - \tan^{- 1} \frac{3}{4})$ is,role="math" localid="1646575560738" $- \frac{48 + 25 \sqrt{3}}{39}$ .

Step by step solution

Step 1 Given expression is,

$\tan (\sin^{- 1} (\frac{- 1}{2}) - \tan^{- 1} \frac{3}{4})$

Consider,

$伪 = \sin^{- 1} (\frac{- 1}{2}) \sin 伪 = \frac{- 1}{2} 尾 = \tan^{- 1} \frac{3}{4} \tan 尾 = \frac{3}{4}$

The formulae used in this problem are,

$\tan (伪 - 尾) = \frac{\tan 伪 - \tan 尾}{1 + \tan 伪 . \tan 尾} \sin^{2} 伪 + \cos^{2} 伪 = 1$

Step 2 Find the value of cos α.

$\cos 伪 = \sqrt{1 - \sin^{2} 伪} = \sqrt{1 - {(\frac{- 1}{2})}^{2}} = \sqrt{1 - \frac{1}{4}} \cos 伪 = \frac{\sqrt{3}}{2}$

Step 3 we know that, tan α=sin αcos α.

Substitute the known value in the formula.

$\tan 伪 = \frac{(\frac{- 1}{2})}{\frac{\sqrt{3}}{2}} \tan 伪 = \frac{- 1}{\sqrt{3}}$

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

$\tan (伪 - 尾) = \frac{(\frac{- 1}{\sqrt{3}}) - \frac{3}{4}}{1 + (\frac{- 1}{\sqrt{3}}) (\frac{3}{4})} = \frac{\frac{- 4 - 3 \sqrt{3}}{4 \sqrt{3}}}{\frac{4 \sqrt{3} - 3}{4 \sqrt{3}}} = \frac{- 4 - 3 \sqrt{3}}{\sqrt{3} (4 - \sqrt{3})}$

Multiply by $\sqrt{3} (4 + \sqrt{3})$ on both the numerator and the denominator.

$\frac{- 4 - 3 \sqrt{3}}{\sqrt{3} (4 - \sqrt{3})} = \frac{- 4 - 3 \sqrt{3}}{\sqrt{3} (4 - \sqrt{3})} (\frac{\sqrt{3} (4 + \sqrt{3})}{\sqrt{3} (4 + \sqrt{3})}) = \frac{- 48 - 25 \sqrt{3}}{39} = - \frac{48 + 25 \sqrt{3}}{39}$ Hence, $\tan (\sin^{- 1} (\frac{- 1}{2}) - \tan^{- 1} \frac{3}{4}) = - \frac{48 + 25 \sqrt{3}}{39}$

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91影视

In Problems $58 - 63$ , find the exact value of each expression.
$\tan (\sin^{- 1} (\frac{- 1}{2}) - \tan^{- 1} \frac{3}{4})$ .

Short Answer

Step by step solution

Step 1 Given expression is,

Step 2 Find the value of cos α.

Step 3 we know that, tan α=sin αcos α.

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

One App. One Place for Learning.

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91影视

In Problems 58-63, find the exact value of each expression.tansin-1-12-tan-134.

Short Answer

Step by step solution

Step 1 Given expression is,

Step 2 Find the value of cos α.

Step 3 we know that, tan α=sin αcos α.

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

One App. One Place for Learning.

Most popular questions from this chapter

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In Problems $58 - 63$ , find the exact value of each expression.
$\tan (\sin^{- 1} (\frac{- 1}{2}) - \tan^{- 1} \frac{3}{4})$ .