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Math  /  Trigonometry

QuestionUse an appropriate Half-Angle Formula to find the exact value of the expression. sin⁡(9π8)\sin \left(\frac{9 \pi}{8}\right)

Studdy Solution

STEP 1

What is this asking? We need to find the exact value of sin⁡(9π8)\sin(\frac{9\pi}{8}) using a half-angle formula. Watch out! Remember to choose the correct half-angle formula based on the quadrant of the angle and be careful with the signs!

STEP 2

1. Find the double angle
2. Choose the appropriate half-angle formula
3. Evaluate the formula

STEP 3

We're given the angle 9π8\frac{9\pi}{8} inside the sine function.
We want to use a half-angle formula, so we need to think of this angle as *half* of some other angle.
Let's call that other angle xx.
So, we have x2=9π8\frac{x}{2} = \frac{9\pi}{8}.

STEP 4

To find xx, we **multiply both sides** of the equation by **2**: 2⋅x2=2⋅9π82 \cdot \frac{x}{2} = 2 \cdot \frac{9\pi}{8} x=18π8=9π4x = \frac{18\pi}{8} = \frac{9\pi}{4}So, our double angle is x=9π4x = \frac{9\pi}{4}.

STEP 5

The half-angle formula for sine is: sin⁡(x2)=±1−cos⁡(x)2\sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos(x)}{2}} We need to figure out whether to use the positive or negative square root.

STEP 6

Our original angle is 9π8\frac{9\pi}{8}.
Since π<9π8<3π2\pi < \frac{9\pi}{8} < \frac{3\pi}{2}, this angle is in the **third quadrant**.
Sine is **negative** in the third quadrant, so we'll use the **negative** square root.

STEP 7

sin⁡(9π8)=−1−cos⁡(9π4)2\sin\left(\frac{9\pi}{8}\right) = -\sqrt{\frac{1 - \cos(\frac{9\pi}{4})}{2}}

STEP 8

We need to find cos⁡(9π4)\cos(\frac{9\pi}{4}).
Since 9π4=2π+π4\frac{9\pi}{4} = 2\pi + \frac{\pi}{4}, it's the same as π4\frac{\pi}{4} which is 4545 degrees.
We know that cos⁡(π4)=22\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}.

STEP 9

Now, we **substitute** this value into our half-angle formula: sin⁡(9π8)=−1−222\sin\left(\frac{9\pi}{8}\right) = -\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} To simplify, we can **multiply** the numerator and denominator inside the square root by **2**: −2−24-\sqrt{\frac{2 - \sqrt{2}}{4}} Since 4=2\sqrt{4} = 2, we can simplify further: −2−22-\frac{\sqrt{2 - \sqrt{2}}}{2}

STEP 10

The exact value of sin⁡(9π8)\sin(\frac{9\pi}{8}) is −2−22-\frac{\sqrt{2 - \sqrt{2}}}{2}.

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