Find the value of $cos(x)$ and $sin(x)$ based on the unit circle

Answer 1

Given $x = \frac{5\pi}{4}$, we need to find $\cos(x)$ and $\sin(x)$ using the unit circle.

Step 1: Locate the angle $\frac{5\pi}{4}$ on the unit circle. This angle is in the third quadrant.

Step 2: Determine the reference angle. The reference angle for $\frac{5\pi}{4}$ is $\frac{\pi}{4}$.

Step 3: Recall the unit circle values for $\frac{\pi}{4}$. The coordinates are $(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}})$ in the third quadrant.

Therefore, $\cos(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}$ and $\sin(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}$.

Answer 2

Given $x = frac{2pi}{3}$, we need to find $cos(x)$ and $sin(x)$ using the unit circle.

Step 1: Locate the angle $frac{2pi}{3}$ on the unit circle. This angle is in the second quadrant.

Step 2: Determine the reference angle. The reference angle for $frac{2pi}{3}$ is $frac{pi}{3}$.

Step 3: Recall the unit circle values for $frac{pi}{3}$. The coordinates are $(-frac{1}{2}, frac{sqrt{3}}{2})$ in the second quadrant.

Therefore, $cos(frac{2pi}{3}) = -frac{1}{2}$ and $sin(frac{2pi}{3}) = frac{sqrt{3}}{2}$.

Answer 3

Given $x = frac{7pi}{6}$, we need to find $cos(x)$ and $sin(x)$ using the unit circle.

The coordinates of $frac{7pi}{6}$ on the unit circle are $(-frac{sqrt{3}}{2}, -frac{1}{2})$.

Therefore, $cos(frac{7pi}{6}) = -frac{sqrt{3}}{2}$ and $sin(frac{7pi}{6}) = -frac{1}{2}$.