9 - 65. your calculator can calculate function in both degrees and radians. see if you can determine how to…

Explanation:

Step1: Calculate $\sin(60^{\circ})$ in degree - mode

We know that $\sin(60^{\circ})=\frac{\sqrt{3}}{2}\approx0.866$.

Step2: Calculate $\sin(\frac{\pi}{3})$ in radian - mode

Since $\frac{\pi}{3}$ radians is equivalent to $60^{\circ}$, and the sine function is periodic and well - defined for both degree and radian measures of angles, $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}\approx0.866$. The answers are the same because $60^{\circ}=\frac{\pi}{3}$ radians, and the sine function is angle - measure independent (as long as we convert correctly between degrees and radians).

Step3: Find $\sin(\frac{\pi}{2})$

We know that $\sin(\frac{\pi}{2}) = 1$. In degree measure, $\frac{\pi}{2}$ radians is $90^{\circ}$, and $\sin(90^{\circ})=1$. Angles that have the same sine as $\sin(\frac{\pi}{2})$ are $90^{\circ}+ 360^{\circ}n$ or $\frac{\pi}{2}+2\pi n$ ($n\in\mathbb{Z}$) in degree and radian measures respectively.

Answer:

a. $\sin(60^{\circ})=\frac{\sqrt{3}}{2}\approx0.866$, $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}\approx0.866$. They are the same because $60^{\circ}=\frac{\pi}{3}$ radians. b. $\sin(\frac{\pi}{2}) = 1$. Angles with the same sine are $\frac{\pi}{2}+2\pi n$ radians or $90^{\circ}+ 360^{\circ}n$ degrees, $n\in\mathbb{Z}$.