analyzing trigonometric functions\nwhat are t...
analyzing trigonometric functions\nwhat are the input and output values for determining the sine of 60°?\ninput: $\frac{2}{sqrt{3}}$; output: 60°\ninput: 60°; output: $\frac{sqrt{3}}{2}$\ninput: 60°; output: $\frac{2}{sqrt{3}}$\ninput: $\frac{sqrt{3}}{2}$; output: 60°
Answer
# Explanation:
## Step1: Recall sine - function definition
The sine of an angle $\theta$ in a right - triangle is defined as $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$.
## Step2: Identify angle and sides
For $\angle B = 60^{\circ}$ in right - triangle $ABC$, the opposite side to $\angle B$ is $AC = 8\sqrt{3}$ and the hypotenuse $AB = 16$.
## Step3: Calculate $\sin60^{\circ}$
$\sin60^{\circ}=\frac{AC}{AB}=\frac{8\sqrt{3}}{16}=\frac{\sqrt{3}}{2}$. In the context of a trigonometric function $y = \sin(x)$, the input $x$ is the angle measure and the output $y$ is the sine value of that angle. So for $\sin60^{\circ}$, the input is $60^{\circ}$ and the output is $\frac{\sqrt{3}}{2}$.
# Answer:
input: $60^{\circ}$; output: $\frac{\sqrt{3}}{2}$