Chapter 7: Q. 110 (page 488)

Show that $\cos \frac{π}{8} = \frac{\sqrt{2 + \sqrt{2}}}{2}$ and use it to find $\sin \frac{π}{16}, \cos \frac{π}{16}$ .

Short Answer

Expert verified

The value of $\sin \frac{π}{16}$ is $\frac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{2}$ .

The value of $\cos \frac{π}{16}$ is $\frac{\sqrt{2 - \sqrt{2 + \sqrt{2}}}}{2}$ .

Step by step solution

Step 1. Given information.

Consider the given question,

$\cos \frac{π}{8} = \frac{\sqrt{2 + \sqrt{2}}}{2}$

Using half-angle formulas,

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

Consider $\cos θ = \cos (\frac{π}{4})$ . Then,

localid="1646421590598" $θ = \frac{π}{4} \frac{θ}{2} = \frac{π}{8}$

Step 2. Use the half angle formula.

We know that $\cos (\frac{π}{4}) = \frac{\sqrt{2}}{2}$ .

Using half-angle formulas,

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

As $\frac{θ}{2}$ lies in the first quadrant,

$\cos (\frac{π}{8}) = \frac{\sqrt{2 + \sqrt{2}}}{2}$

Step 3. Use the trigonometric identities.

Consider $\cos θ = \cos (\frac{π}{8})$ . Then,

role="math" localid="1646421897677" $θ = \frac{π}{8} \frac{θ}{2} = \frac{π}{16}$

As $\frac{θ}{2}$ lies in the first quadrant,

$\cos (\frac{π}{8}) = \frac{\sqrt{2 + \sqrt{2}}}{2}$

Using half-angle formulas,

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

Step 4. Solve for sinπ16.

Consider the given question,

$\sin \frac{θ}{2} = \pm \sqrt{\frac{1 - \cos θ}{2}} \sin \frac{θ}{2} = \pm \sqrt{\frac{1 - (\frac{\sqrt{2 + \sqrt{2}}}{2})}{2}} \sin \frac{θ}{2} = \pm \frac{\sqrt{2 - \sqrt{2 + \sqrt{2}}}}{2}$

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Show that $\cos \frac{π}{8} = \frac{\sqrt{2 + \sqrt{2}}}{2}$ and use it to find $\sin \frac{π}{16}, \cos \frac{π}{16}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the half angle formula.

Step 3. Use the trigonometric identities.

Step 4. Solve for sinπ16.

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Show thatcosπ8=2+22and use it to findsinπ16,cosπ16.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the half angle formula.

Step 3. Use the trigonometric identities.

Step 4. Solve for sinπ16.

One App. One Place for Learning.

Most popular questions from this chapter

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Show that $\cos \frac{π}{8} = \frac{\sqrt{2 + \sqrt{2}}}{2}$ and use it to find $\sin \frac{π}{16}, \cos \frac{π}{16}$ .