Q. 22 In Problems 19鈥�28, use the Hal... [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. 22 (page 495)

In Problems 19鈥�28, use the Half-angle Formulas to find the exact value of each expression.
$\tan \frac{9 蟺}{8}$

Short Answer

Expert verified

The exact value of the expression $\tan \frac{9 蟺}{8}$ by using Half angle formula is $\sqrt{3 - 2 \sqrt{2}}$

Step by step solution

Step 1. Given

An expression $\tan \frac{9 蟺}{8}$

To find the exact value of the expression

Step 2. Half angle formula

The half angle formula is,

$\tan \frac{伪}{2} = 卤 \sqrt{\frac{1 - \cos 伪}{1 + \cos 伪}}$

Step 3. Consider an angle

Consider $\tan 伪 = \frac{9 蟺}{4}$

$伪 = \frac{9 蟺}{4} \frac{伪}{2} = \frac{9 蟺}{8}$

Step 4. Compute cos α

$\cos \frac{9 蟺}{4} = \cos (2 蟺 + \frac{蟺}{4}) = \cos \frac{蟺}{4} = \frac{\sqrt{2}}{2}$

Step 5. Use the half-angle formula

By Half-angle formula,

$\tan \frac{伪}{2} = 卤 \sqrt{\frac{1 - \cos 伪}{1 + \cos 伪}} \tan \frac{\frac{9 蟺}{4}}{2} = 卤 \sqrt{\frac{1 - \cos \frac{9 蟺}{4}}{1 + \cos \frac{9 蟺}{4}}} = 卤 \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \frac{\sqrt{2}}{2}}} = 卤 \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}}$

Step 6. Take conjugation and solve

$\tan \frac{9 蟺}{8} = 卤 \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}} = 卤 \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}} 脳 \sqrt{\frac{2 - \sqrt{2}}{2 - \sqrt{2}}} = 卤 \sqrt{\frac{{(2 - \sqrt{2})}^{2}}{2^{2} - {(\sqrt{2})}^{2}}} = 卤 \sqrt{\frac{4 - 4 \sqrt{2} + 2}{4 - 2}} = 卤 \sqrt{\frac{6 - 4 \sqrt{2}}{2}} = 卤 \sqrt{3 - 2 \sqrt{2}}$

Step 7. Find the exact value

Since $\frac{9 蟺}{8}$ lies in third quadrant,

$\tan \frac{9 蟺}{8} = \sqrt{3 - 2 \sqrt{2}}$

The exact value of the expression is $\sqrt{3 - 2 \sqrt{2}}$

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91影视

In Problems 19鈥�28, use the Half-angle Formulas to find the exact value of each expression.
$\tan \frac{9 蟺}{8}$

Short Answer

Step by step solution

Step 1. Given

Step 2. Half angle formula

Step 3. Consider an angle

Step 4. Compute cos α

Step 5. Use the half-angle formula

Step 6. Take conjugation and solve

Step 7. Find the exact value

One App. One Place for Learning.

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91影视

In Problems 19鈥�28, use the Half-angle Formulas to find the exact value of each expression.tan9蟺8

Short Answer

Step by step solution

Step 1. Given

Step 2. Half angle formula

Step 3. Consider an angle

Step 4. Compute cos α

Step 5. Use the half-angle formula

Step 6. Take conjugation and solve

Step 7. Find the exact value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Discrete Mathematics

Probability and Statistics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems 19鈥�28, use the Half-angle Formulas to find the exact value of each expression.
$\tan \frac{9 蟺}{8}$