Q. 26 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. 26 (page 495)

In Problems 19鈥�28, use the Half-angle Formulas to find the exact value of each expression.
$c s c \frac{7 蟺}{8}$

Short Answer

Expert verified

The exact value of the expression $c s c \frac{7 蟺}{8}$ by using Half angle formula is $\sqrt{4 + 2 \sqrt{2}}$

Step by step solution

Step 1. Given

An expression $c s c \frac{7 蟺}{8}$

To find the exact value of the expression

Step 2. Half angle formula

We know that

c s c 胃 = \frac{1}{\sin 胃} c s c \frac{7 蟺}{8} = \frac{1}{\sin \frac{7 蟺}{8}}

The half angle formula is,

$\sin \frac{胃}{2} = 卤 \sqrt{\frac{1 - \cos 胃}{2}}$

Step 3. Consider an angle

Consider

$c s c \frac{7 蟺}{4} = c s c 胃胃 = \frac{7 蟺}{4} \frac{胃}{2} = \frac{\frac{7 蟺}{4}}{2} = \frac{7 蟺}{8}$

Step 4. Use the half-angle formula

By Half-angle formula,

$\sin \frac{7 蟺}{4} = 卤 \sqrt{\frac{1 - \cos \frac{7 蟺}{4}}{2}}$

We know that,

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

Step 5. Substitute the values

Substitute the values in the above formula,

$\sin \frac{7 蟺}{8} = 卤 \sqrt{\frac{1 - \cos \frac{7 蟺}{4}}{2}} = 卤 \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = 卤 \sqrt{\frac{\frac{2 - \sqrt{2}}{2}}{2}} = 卤 \sqrt{\frac{2 - \sqrt{2}}{4}} = 卤 \frac{\sqrt{2 - \sqrt{2}}}{2}$

Step 6. Find the exact value.

Since $\frac{7 蟺}{8}$ lies in the second quadrant,

$\sin \frac{7 蟺}{8} = \frac{\sqrt{2 - \sqrt{2}}}{2}$

So, $c s c \frac{7 蟺}{8} = \frac{2}{\sqrt{2 - \sqrt{2}}}$

Step 7. Take conjugate and solve the sum

By conjugation,

$\frac{2}{\sqrt{2 - \sqrt{2}}} = \frac{2}{\sqrt{2 - \sqrt{2}}} 脳 \frac{\sqrt{2 + \sqrt{2}}}{\sqrt{2 + \sqrt{2}}} = \frac{2 \sqrt{2 + \sqrt{2}}}{\sqrt{2^{2} - \sqrt{2^{2}}}} = \frac{2 \sqrt{2 + \sqrt{2}}}{\sqrt{4 - 2}} = \frac{2 \sqrt{2 + \sqrt{2}}}{\sqrt{2}} = \sqrt{2 .} \sqrt{2 + \sqrt{2}} = \sqrt{4 + 2 \sqrt{2}}$

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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.
$c s c \frac{7 蟺}{8}$

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. Substitute the values

Step 6. Find the exact value.

Step 7. Take conjugate and solve the sum

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.csc7蟺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. Substitute the values

Step 6. Find the exact value.

Step 7. Take conjugate and solve the sum

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Pure Maths

Decision Maths

Probability and Statistics

Discrete Mathematics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 19鈥�28, use the Half-angle Formulas to find the exact value of each expression.
$c s c \frac{7 蟺}{8}$