Chapter 7: Q. 21 (page 495)

In Problems 19–28, use the Half-angle Formulas to find the exact value of each expression.
$\tan \frac{7 π}{8}$

Short Answer

Expert verified

The exact value of the expression $\tan \frac{7 π}{8}$ by using Half angle formula is $1 - \sqrt{2}$

Step by step solution

Step 1. Given

An expression $\tan \frac{7 π}{8}$

To find the exact value of the expression

Step 2. Half angle formula

The half angle formula is,

$\tan \frac{α}{2} = \pm \sqrt{\frac{1 - \cos α}{1 + \cos α}}$

Step 3. Consider an angle

Consider,

$\tan α = \tan \frac{7 π}{8} α = \frac{7 π}{8} \frac{α}{2} = \frac{\frac{7 π}{4}}{2}$

Let $α = \frac{7 π}{4}$

We know that, $\tan \frac{α}{2} = \frac{1 - \cos α}{\sin α}$

Step 4. Use the half-angle formula

By Half-angle formula,

$\tan \frac{α}{2} = \frac{1 - \cos α}{\sin α}$

Compute for $\cos α, \sin α$

$\cos \frac{7 π}{4} = \cos (2 π - \frac{π}{4}) = \cos (- \frac{π}{4}) = \cos \frac{π}{4} = \frac{\sqrt{2}}{2}$

$\sin \frac{7 π}{4} = \sin (2 π - \frac{π}{4}) = \sin (- \frac{π}{4}) = - \sin \frac{π}{4} = - \frac{\sqrt{2}}{2}$

Step 5. Find the exact value

Since $\tan \frac{α}{2} = \frac{1 - \cos α}{\sin α}$

$\tan \frac{\frac{7 π}{4}}{2} = \frac{1 - \frac{\sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} = \frac{\frac{2 - \sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} = - (\frac{2 - \sqrt{2}}{\sqrt{2}}) = 1 - \sqrt{2}$

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In Problems 19–28, use the Half-angle Formulas to find the exact value of each expression.
$\tan \frac{7 π}{8}$

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.tan7π8

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

Theoretical and Mathematical Physics

Calculus

Geometry

Applied Mathematics

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 19–28, use the Half-angle Formulas to find the exact value of each expression.
$\tan \frac{7 π}{8}$