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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 16. (page 457)

In Problems 9鈥�36, find the exact value of each expression.
$\sec (\tan^{- 1} \sqrt{3})$

Short Answer

Expert verified

The value of the expression $\sec (\tan^{- 1} \sqrt{3}) = 2$ .

Step by step solution

01

Step 1. Given information.

The given expression is:

$\sec (\tan^{- 1} \sqrt{3})$

Let's take $胃 = \tan^{- 1} \sqrt{3}$ ,

$\tan 胃 = \sqrt{3}$

$- \frac{蟺}{2} 鈮� 胃鈮� \frac{蟺}{2}$ is the domain of the function.

02

Step 2. Find the value of θ.

Initially, we have function $\tan^{- 1}$ and the bounds of tan function are going to be in the interval $[- \frac{蟺}{2}, \frac{蟺}{2}]$ . It represents the range of $\tan^{- 1}$ .

By using the unit circle, we can say that $胃$ must be equal to $\frac{蟺}{3}$ , so that we can get $\sqrt{3}$ .

$\tan 胃 = \sqrt{3} \tan^{- 1} \sqrt{3} = \frac{蟺}{3}$

03

Step 3. Find the exact value of the expression.

$\sec (\tan^{- 1} \sqrt{3}) = \sec (\frac{蟺}{3}) \sec (胃) = \frac{1}{\cos 胃} \sec (\frac{蟺}{3}) = \frac{1}{\cos (\frac{蟺}{3})}$

By using the unit circle, we know that $\cos (\frac{蟺}{3}) = \frac{1}{2}$ , then

$\sec (\frac{蟺}{3}) = \frac{1}{\frac{1}{2}} = 2$

Therefore, $\sec (\tan^{- 1} \sqrt{3}) = 2$ .

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Most popular questions from this chapter

Establish each identity.

$\frac{1 + \sin 胃 + \cos 胃}{1 + \sin 胃 - \cos 胃} = \frac{1 + \cos 胃}{\sin 胃}$

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.

$\frac{1}{1 - \sin 胃} + \frac{1}{1 + \sin 胃} = 2 s e c^{2} 胃$

If $\tan 伪 = x + 1$ and role="math" localid="1646425111199" $\tan 尾 = x - 1,$ show that

role="math" localid="1646426027733" $2 c o t (伪 - b) = x^{2}$

Express the product $\sin (4 胃) \sin (2 胃)$ as a sum containing only sines or only cosines.

See all solutions

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