Find the exact values of $sin( heta)$ and $cos( heta)$ for $ heta = frac{3pi}{4}$ using the unit circle.
Answer 1
Ava Martin
To find the exact values of $\sin(\theta)$ and $\cos(\theta)$ for $\theta = \frac{3\pi}{4}$, we can use the unit circle.
First, note that $\theta = \frac{3\pi}{4}$ is in the second quadrant. In the unit circle, the angle $\frac{3\pi}{4}$ corresponds to $135^\circ$.
For angles in the second quadrant, the sine value is positive and the cosine value is negative. The reference angle for $\frac{3\pi}{4}$ is $\frac{\pi}{4}$ (or $45^\circ$), where $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ and $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.
Therefore, $\sin(\frac{3\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ and $\cos(\frac{3\pi}{4}) = -\cos(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$.
So, the exact values are $\sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2}$ and $\cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2}$.
Answer 2
Christopher Garcia
To determine $sin( heta)$ and $cos( heta)$ for $ heta = frac{3pi}{4}$ using the unit circle:
1. Recognize that $ heta = frac{3pi}{4}$ falls in the second quadrant.
2. In the second quadrant, $sin( heta)$ remains positive, and $cos( heta)$ is negative.
3. The reference angle is $frac{pi}{4}$, with known values $sin(frac{pi}{4}) = frac{sqrt{2}}{2}$ and $cos(frac{pi}{4}) = frac{sqrt{2}}{2}$.
4. Thus, $sin(frac{3pi}{4}) = frac{sqrt{2}}{2}$ and $cos(frac{3pi}{4}) = -frac{sqrt{2}}{2}$.
Therefore, $sin(frac{3pi}{4}) = frac{sqrt{2}}{2}$, $cos(frac{3pi}{4}) = -frac{sqrt{2}}{2}$.
Answer 3
Sophia Williams
Using the unit circle for $ heta = frac{3pi}{4}$:
1. The angle is in the second quadrant.
2. Reference angle is $frac{pi}{4}$.
3. Hence, $sin(frac{3pi}{4}) = frac{sqrt{2}}{2}$ and $cos(frac{3pi}{4}) = -frac{sqrt{2}}{2}$.
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