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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 495)

In Problems 19鈥�28, use the Half-angle Formulas to find the exact value of each expression.
$\cos 165^{0}$

Short Answer

Expert verified

The exact value of the expression $\cos 165^{0}$ by using Half angle formula is $- \frac{\sqrt{2 + \sqrt{3}}}{2}$

Step by step solution

01

Step 1. Given

An expression $\cos 165^{0}$

To find the exact value of the expression

02

Step 2. Half angle formula

The half angle formula is,

$\cos \frac{胃}{2} = \sqrt{\frac{1 + \cos 胃}{2}}$

03

Step 3. Consider an angle

Consider,

$\cos 胃 = \cos 330 掳 \frac{胃}{2} = \frac{330 掳}{2} = 165 掳$

04

Step 4. Use the half-angle formula

By Half-angle formula,

$\cos 165 掳 = 卤 \sqrt{\frac{1 + \cos 330 掳}{2}}$

We know that, $\cos 330 掳 = \cos (360 掳 - 30 掳) = \cos 30 掳 = \frac{\sqrt{3}}{2}$

05

Step 5. Find the exact value

$\cos 165 掳 = 卤 \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = 卤 \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}} = 卤 \sqrt{\frac{2 + \sqrt{3}}{4}} = 卤 \frac{\sqrt{2 + \sqrt{3}}}{2}$

06

Step 6. Approximate the value with quadrant

Since $\cos 165 掳$ lies in second quadrant,

$\cos 165 掳 = - \frac{\sqrt{2 + \sqrt{3}}}{2}$

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Most popular questions from this chapter

Show that $\tan^{- 1} v + c o t^{- 1} v = \frac{蟺}{2}$

Express the product $\sin (3 胃) \sin (5 胃)$ as a sum containing only sines or only cosines.

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

True or False

$\tan (2 胃) = \frac{2 \tan 胃}{1 - \tan^{2} 胃}$

Suppose that f and g are two functions with the same domain. If $f (x) = g (x)$ for every x in the domain, the equation is called a(n) ____. Otherwise, it is called a(n) _____ equation.

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