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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 23. (page 457)

In Problems 9鈥�36, find the exact value of each expression.
$\sin^{- 1} (\cos (- \frac{7 蟺}{6}))$

Short Answer

Expert verified

The value of the expression $\sin^{- 1} (\cos (- \frac{7 蟺}{6})) = - \frac{蟺}{3}$ .

Step by step solution

01

Step 1. Write the given information.

The given expression is:

$\sin^{- 1} (\cos (- \frac{7 蟺}{6}))$

02

Use the trigonometric identities.

Use the identity $\cos (- x) = - \cos x$ ,

localid="1646471658080" $\sin^{- 1} [\cos (- \frac{7 蟺}{6})] = \sin^{- 1} [- \cos (\frac{7 蟺}{6})] = \sin^{- 1} [- \cos (蟺 + \frac{蟺}{6})]$

Use the identity $\cos (蟺 + x) = \cos x$ ,

$\sin^{- 1} [- \cos (\frac{蟺}{6})] = \sin^{- 1} (- \frac{\sqrt{3}}{2})$

03

Step 3. Find the exact value of the expression.

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

Let $胃 = \sin^{- 1} (- \frac{\sqrt{3}}{2})$

$\sin 胃 = (- \frac{\sqrt{3}}{2})$

The only angle $胃$ in the interval $[- \frac{蟺}{2}, \frac{蟺}{2}]$ , whose sine value is $- \frac{\sqrt{3}}{2}$ is $- \frac{蟺}{3}$ .

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

Therefore, $\sin^{- 1} (\cos (- \frac{7 蟺}{6})) = - \frac{蟺}{3}$ .

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Most popular questions from this chapter

True or False: $\sin (- 胃) + \cos (- 胃) = \cos (胃) - \sin (胃)$

In Problems 9鈥�36, find the exact value of each expression.

$\cos^{- 1} (\sin \frac{5 蟺}{4})$

Sawtooth Curve An oscilloscope often displays a sawtooth curve. This curve can be approximated by sinusoidal curves of varying periods and amplitudes. A first approximation to the sawtooth curve is given by

$y = \frac{1}{2} \sin (2 蟺 x) + \frac{1}{4} \sin (4 蟺 x)$

Show that $y = \sin (2 蟺 x) \cos^{2} (蟺 x)$

Establish each identity.

$\frac{1 + \sin 胃}{1 - \sin 胃} - \frac{1 - \sin 胃}{1 + \sin 胃} = 4 \tan 胃 s e c 胃$

Establish each identity.

$\frac{s e c 胃}{1 - \sin 胃} = \frac{1 + \sin 胃}{\cos^{3} 胃}$

See all solutions

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