Question

without using a calculator, compute the sine and cosine of $\frac{3\pi}{4}$ by using the reference angle.
what is the reference angle? radians.
in what quadrant is this angle? (answer 1, 2, 3, or 4)
$\sin(\frac{3\pi}{4}) = $
$\cos(\frac{3\pi}{4}) = $
(type sqrt(2) for $\sqrt{2}$ and sqrt(3) for $\sqrt{3}$.)
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Explanation:

Step1: Determine the reference angle

The angle \(\frac{3\pi}{4}\) is in the second quadrant. The formula for the reference angle \(\theta'\) of an angle \(\theta\) in the second quadrant is \(\theta'=\pi - \theta\).
So, \(\theta'=\pi-\frac{3\pi}{4}=\frac{\pi}{4}\) radians.

Step2: Determine the quadrant

Since \(\frac{\pi}{2}<\frac{3\pi}{4}<\pi\), the angle \(\frac{3\pi}{4}\) is in the second quadrant.

Step3: Find \(\sin(\frac{3\pi}{4})\)

We know that \(\sin(\theta)=\sin(\theta')\) in the second quadrant. Since \(\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\), so \(\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}\) (or \(sqrt(2)/2\)).

Step4: Find \(\cos(\frac{3\pi}{4})\)

We know that \(\cos(\theta)=-\cos(\theta')\) in the second quadrant. Since \(\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\), so \(\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}\) (or \(-sqrt(2)/2\)).

Answer:

Reference angle: \(\frac{\pi}{4}\) radians.
Quadrant: \(2\)
\(\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}\)
\(\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}\)