Chapter 7: Q. 20 (page 495)

In Problems 19–28, use the Half-angle Formulas to find the exact value of each expression.
$\cos 22.5 °$

Short Answer

Expert verified

The exact value of the expression $\cos 22.5 °$ by using Half angle formula is $\frac{\sqrt{2 + \sqrt{2}}}{2}$

Step by step solution

Step 1. Given

An expression $\cos 22.5 °$

To find the exact value of the expression

Step 2. Half angle formula

The half angle formula is,

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

Step 3. Consider an angle

Consider

$\cos θ = \cos 45 ° θ = 45 ° \frac{θ}{2} = \frac{45 °}{2} = 22.5 °$

Step 4. Use the half-angle formula

By Half-angle formula,

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

$\cos 22.5 ° = \sqrt{\frac{1 + \cos 45 °}{2}}$

We know that, $\cos 45 ° = \frac{\sqrt{2}}{2}$ ,

role="math" localid="1646525507945" $\cos 22.5 ° = \pm \sqrt{\frac{1 + (\frac{\sqrt{2}}{2})}{2}} = \pm \sqrt{\frac{2 + \sqrt{2}}{4}} = \pm \frac{\sqrt{2 + \sqrt{2}}}{2}$

Step 5. Find the exact value

Since $22.5 °$ lies in the first quadrant,

$\cos 22.5 ° = \frac{\sqrt{2 + \sqrt{2}}}{2}$

The exact value of the expression is $\frac{\sqrt{2 + \sqrt{2}}}{2}$

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In Problems 19–28, use the Half-angle Formulas to find the exact value of each expression.
$\cos 22.5 °$

Short Answer

Step by step solution

Step 1. Given

Step 2. Half angle formula

Step 3. Consider an angle

Step 4. Use the half-angle formula

Step 5. Find the exact value

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In Problems 19–28, use the Half-angle Formulas to find the exact value of each expression.cos22.5°

Short Answer

Step by step solution

Step 1. Given

Step 2. Half angle formula

Step 3. Consider an angle

Step 4. Use the half-angle formula

Step 5. Find the exact value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Mechanics Maths

Logic and Functions

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

In Problems 19–28, use the Half-angle Formulas to find the exact value of each expression.
$\cos 22.5 °$