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Math  /  Trigonometry

QuestionC. sin⁡π8\sin \frac{\pi}{8}

Studdy Solution

STEP 1

1. We need to find the exact value of sin⁡π8 \sin \frac{\pi}{8} .
2. We will use trigonometric identities and known angle values to find this value.
3. The half-angle identity for sine will be useful in this calculation.

STEP 2

1. Identify the relevant trigonometric identity.
2. Apply the half-angle identity for sine.
3. Simplify the expression to find the exact value.

STEP 3

Identify the relevant trigonometric identity. We will use the half-angle identity for sine:
sin⁡θ2=±1−cos⁡θ2\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}
For our problem, θ=π4\theta = \frac{\pi}{4}, since π8=π4×12\frac{\pi}{8} = \frac{\pi}{4} \times \frac{1}{2}.

STEP 4

Apply the half-angle identity for sine. Substitute θ=π4\theta = \frac{\pi}{4} into the identity:
sin⁡π8=1−cos⁡π42\sin \frac{\pi}{8} = \sqrt{\frac{1 - \cos \frac{\pi}{4}}{2}}

STEP 5

We need the value of cos⁡π4\cos \frac{\pi}{4}. From known trigonometric values:
cos⁡π4=22\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}
Substitute this into the expression:
sin⁡π8=1−222\sin \frac{\pi}{8} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}}

STEP 6

Simplify the expression:
sin⁡π8=24−24\sin \frac{\pi}{8} = \sqrt{\frac{2}{4} - \frac{\sqrt{2}}{4}}
=2−24= \sqrt{\frac{2 - \sqrt{2}}{4}}
=2−22= \frac{\sqrt{2 - \sqrt{2}}}{2}
This is the exact value of sin⁡π8\sin \frac{\pi}{8}.
The exact value of sin⁡π8\sin \frac{\pi}{8} is 2−22\boxed{\frac{\sqrt{2 - \sqrt{2}}}{2}}.

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