Q. 109 Use the fact thatcos蟺12=146+2to... [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. 109 (page 498)

Use the fact that
$\cos \frac{蟺}{12} = \frac{1}{4} (\sqrt{6} + \sqrt{2})$
to find $\sin \frac{蟺}{24}, \cos \frac{蟺}{24}$ .

Short Answer

Expert verified

The value of $\sin \frac{蟺}{24}$ is $\frac{\sqrt{2}}{4} \sqrt{4 - \sqrt{6} + \sqrt{2}}$ .

The value of $\cos \frac{蟺}{24}$ is $\frac{\sqrt{2}}{4} \sqrt{4 + \sqrt{6} + \sqrt{2}}$ .

Step by step solution

Step 1. Given information.

Consider the given question,

$\cos \frac{蟺}{12} = \frac{1}{4} (\sqrt{6} + \sqrt{2})$

Assume $\sin \frac{蟺}{24}$ to be $\sin 胃$ . Then,

$\sin \frac{蟺}{24} = \sin 胃胃 = \frac{蟺}{24}$

Also, $\cos \frac{蟺}{12} = \cos (2 脳 \frac{蟺}{24}) = \cos (2 胃)$

Step 2. Use the half angle formula.

Using the half angle formula,

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

Substitute the value of $\cos (2 胃)$ in the equation,

role="math" localid="1646420828601" $\sin^{2} \frac{蟺}{24} = \frac{1 - \cos (\frac{蟺}{12})}{2} \sin \frac{蟺}{24} = \sqrt{\frac{1 - \cos (\frac{蟺}{12})}{2}} \sin \frac{蟺}{24} = \sqrt{\frac{1 - (\frac{\sqrt{6} + \sqrt{2}}{4})}{2}}$

Step 3. Simplify the expression on the right side.

Continuing the above equation,

$\sin \frac{蟺}{24} = \sqrt{\frac{4 - (\frac{\sqrt{6} + \sqrt{2}}{4})}{1}} \sin \frac{蟺}{24} = \sqrt{\frac{4 - (\sqrt{6} + \sqrt{2})}{8}} \sin \frac{蟺}{24} = \frac{1}{2 \sqrt{2}} \sqrt{4 - \sqrt{6} + \sqrt{2}} \sin \frac{蟺}{24} = \frac{\sqrt{2}}{4} \sqrt{4 - \sqrt{6} + \sqrt{2}}$

Step 4. Use the half angle formula.

Using the half angle formula,

$\cos^{2} 胃 = \frac{1 + \cos (2 胃)}{2}$

Substitute the value of $\cos (2 胃)$ in the equation,

$\cos^{2} \frac{蟺}{24} = \frac{1 + \cos (\frac{蟺}{12})}{2} \cos \frac{蟺}{24} = \sqrt{\frac{1 + (\frac{\sqrt{6} + \sqrt{2}}{4})}{2}} \cos \frac{蟺}{24} = \sqrt{\frac{4 + (\frac{\sqrt{6} + \sqrt{2}}{4})}{2}} \cos \frac{蟺}{24} = \sqrt{4 + \sqrt{6} + \sqrt{2}} \cos \frac{蟺}{24} = \frac{\sqrt{2}}{4} \sqrt{4 + \sqrt{6} + \sqrt{2}}$

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91影视

Use the fact that
$\cos \frac{蟺}{12} = \frac{1}{4} (\sqrt{6} + \sqrt{2})$
to find $\sin \frac{蟺}{24}, \cos \frac{蟺}{24}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the half angle formula.

Step 3. Simplify the expression on the right side.

Step 4. Use the half angle formula.

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91影视

Use the fact thatcos蟺12=146+2to findsin蟺24,cos蟺24.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the half angle formula.

Step 3. Simplify the expression on the right side.

Step 4. Use the half angle formula.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Pure Maths

Discrete Mathematics

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Use the fact that
$\cos \frac{蟺}{12} = \frac{1}{4} (\sqrt{6} + \sqrt{2})$
to find $\sin \frac{蟺}{24}, \cos \frac{蟺}{24}$ .