Question

analyzing trigonometric functions
what are the input and output values for determining the sine of 60°?
input: $\frac{2}{sqrt{3}}$; output: 60°
input: 60°; output: $\frac{sqrt{3}}{2}$
input: 60°; output: $\frac{2}{sqrt{3}}$

Explanation:

Step1: Recall sine - function definition

The sine function is defined as $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$ in a right - triangle. For an angle $\theta$ in a right - triangle, the input of the sine function is the measure of the angle, and the output is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Step2: Identify angle and ratio for $\sin60^{\circ}$

In a 30 - 60 - 90 triangle, if the angle $\theta = 60^{\circ}$, and considering the side lengths of a 30 - 60 - 90 triangle where the hypotenuse $c$, the side opposite the 30 - degree angle $a$, and the side opposite the 60 - degree angle $b$ with the ratio $a:b:c = 1:\sqrt{3}:2$. For $\sin60^{\circ}$, the input is the angle measure, which is $60^{\circ}$, and the output is $\frac{\text{opposite}}{\text{hypotenuse}}=\frac{\sqrt{3}}{2}$.

Answer:

Input: $60^{\circ}$; output: $\frac{\sqrt{3}}{2}$