Chapter 7: Q. 109 (page 498)

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

Short Answer

Expert verified

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

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

Step by step solution

Step 1. Given information.

Consider the given question,

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

Assume $\sin \frac{π}{24}$ to be $\sin θ$ . Then,

$\sin \frac{π}{24} = \sin θ θ = \frac{π}{24}$

Also, $\cos \frac{π}{12} = \cos (2 \times \frac{π}{24}) = \cos (2 θ)$

Step 2. Use the half angle formula.

Using the half angle formula,

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

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

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

Step 3. Simplify the expression on the right side.

Continuing the above equation,

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

Step 4. Use the half angle formula.

Using the half angle formula,

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

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

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

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

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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Use the fact thatcosπ12=146+2to findsinπ24,cosπ24.

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

Discrete Mathematics

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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