Question

use the information on the circle to match the correct value with each trigonometric expression. you may use a value more than once or not at all

sin \frac{5\pi}{3}

cos \frac{3\pi}{2}

sin 2\pi

sin \frac{7\pi}{4}

cos \frac{7\pi}{6}

sin \frac{3\pi}{4}

values: 0, -\frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}

Explanation:

Step1: Evaluate $\sin\frac{5\pi}{3}$

$\frac{5\pi}{3}$ is in the fourth quadrant, reference angle $\frac{\pi}{3}$. $\sin\frac{5\pi}{3}=-\sin\frac{\pi}{3}=-\frac{\sqrt{3}}{2}$

Step2: Evaluate $\cos\frac{3\pi}{2}$

$\frac{3\pi}{2}$ is on the negative y - axis, $\cos\frac{3\pi}{2}=0$

Step3: Evaluate $\sin2\pi$

$2\pi$ is a full rotation, $\sin2\pi = 0$

Step4: Evaluate $\sin\frac{7\pi}{4}$

$\frac{7\pi}{4}$ is in the fourth quadrant, reference angle $\frac{\pi}{4}$. $\sin\frac{7\pi}{4}=-\sin\frac{\pi}{4}=-\frac{\sqrt{2}}{2}$

Step5: Evaluate $\cos\frac{7\pi}{6}$

$\frac{7\pi}{6}$ is in the third quadrant, reference angle $\frac{\pi}{6}$. $\cos\frac{7\pi}{6}=-\cos\frac{\pi}{6}=-\frac{\sqrt{3}}{2}$

Step6: Evaluate $\sin\frac{3\pi}{4}$

$\frac{3\pi}{4}$ is in the second quadrant, reference angle $\frac{\pi}{4}$. $\sin\frac{3\pi}{4}=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$

Now we can match:

• $\sin\frac{5\pi}{3}$ matches with $-\frac{\sqrt{3}}{2}$
• $\cos\frac{3\pi}{2}$ matches with $0$
• $\sin2\pi$ matches with $0$
• $\sin\frac{7\pi}{4}$ matches with $-\frac{\sqrt{2}}{2}$
• $\cos\frac{7\pi}{6}$ matches with $-\frac{\sqrt{3}}{2}$
• $\sin\frac{3\pi}{4}$ matches with $\frac{\sqrt{2}}{2}$

Answer:

• $\sin\frac{5\pi}{3}$: $-\frac{\sqrt{3}}{2}$
• $\cos\frac{3\pi}{2}$: $0$
• $\sin2\pi$: $0$
• $\sin\frac{7\pi}{4}$: $-\frac{\sqrt{2}}{2}$
• $\cos\frac{7\pi}{6}$: $-\frac{\sqrt{3}}{2}$
• $\sin\frac{3\pi}{4}$: $\frac{\sqrt{2}}{2}$