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Question Evaluate sin⁡(−3π/8)\sin(-3\pi/8) using half-angle formula. Find measure of angle α\alpha where α/2=−3π/8\alpha/2=-3\pi/8. Determine quadrant of half-angle α/2\alpha/2.

Studdy Solution

STEP 1

Assumptions
1. We are given the angle α2=−3π8\frac{\alpha}{2} = -\frac{3\pi}{8}.
2. We need to find sin⁡(−3π8)\sin\left(-\frac{3\pi}{8}\right) using a half-angle formula.
3. The half-angle formula for sine is sin⁡(θ2)=±1−cos⁡(θ)2\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}.
4. The sign in front of the square root depends on the quadrant in which the half-angle lies.
5. We are also given that α=−3π4\alpha = -\frac{3\pi}{4}.

STEP 2

First, we need to determine the quadrant in which the half-angle α2\frac{\alpha}{2} lies. Since α=−3π4\alpha = -\frac{3\pi}{4}, α2=−3π8\frac{\alpha}{2} = -\frac{3\pi}{8} lies in the same quadrant as α\alpha because it is half of α\alpha.

STEP 3

Determine the quadrant of α\alpha. Since α=−3π4\alpha = -\frac{3\pi}{4}, it lies in the third quadrant where both sine and cosine are negative.

STEP 4

Since α2\frac{\alpha}{2} lies in the third quadrant, sin⁡(−3π8)\sin\left(-\frac{3\pi}{8}\right) will be negative because sine is negative in the third quadrant.

STEP 5

Now, we will use the half-angle formula for sine to find sin⁡(−3π8)\sin\left(-\frac{3\pi}{8}\right).
sin⁡(−3π8)=−1−cos⁡(−3π4)2\sin\left(-\frac{3\pi}{8}\right) = -\sqrt{\frac{1 - \cos\left(-\frac{3\pi}{4}\right)}{2}}

STEP 6

We need to find cos⁡(−3π4)\cos\left(-\frac{3\pi}{4}\right). Since cosine is an even function, cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta).

STEP 7

Therefore, cos⁡(−3π4)=cos⁡(3π4)\cos\left(-\frac{3\pi}{4}\right) = \cos\left(\frac{3\pi}{4}\right).

STEP 8

The cosine of 3π4\frac{3\pi}{4} is known to be −22-\frac{\sqrt{2}}{2} because it lies in the second quadrant where cosine is negative.

STEP 9

Substitute the value of cos⁡(3π4)\cos\left(\frac{3\pi}{4}\right) into the half-angle formula.
sin⁡(−3π8)=−1−(−22)2\sin\left(-\frac{3\pi}{8}\right) = -\sqrt{\frac{1 - \left(-\frac{\sqrt{2}}{2}\right)}{2}}

STEP 10

Simplify the expression inside the square root.
sin⁡(−3π8)=−1+222\sin\left(-\frac{3\pi}{8}\right) = -\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}

STEP 11

Combine the terms under the square root.
sin⁡(−3π8)=−2+24\sin\left(-\frac{3\pi}{8}\right) = -\sqrt{\frac{2 + \sqrt{2}}{4}}

STEP 12

Simplify the square root.
sin⁡(−3π8)=−2+24=−2+22\sin\left(-\frac{3\pi}{8}\right) = -\sqrt{\frac{2 + \sqrt{2}}{4}} = -\frac{\sqrt{2 + \sqrt{2}}}{2}
The exact value of sin⁡(−3π8)\sin\left(-\frac{3\pi}{8}\right) is −2+22-\frac{\sqrt{2 + \sqrt{2}}}{2}.
b. The measure of angle α\alpha is given as α=−3π4\alpha = -\frac{3\pi}{4}.
c. The terminal side of the half-angle α2\frac{\alpha}{2} lies in Quadrant III, as determined in steps 2 and 3.

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