What are the input and output values for determining the sine of $60^{\circ}$?
A. input: $\frac{2}{\sqrt{3}}$; output: $60^{\circ}$
B. input: $60^{\circ}$; output: $\frac{\sqrt{3}}{2}$
C. input: $60^{\circ}$; output: $\frac{2}{\sqrt{3}}$
D. input: $\frac{\sqrt{3}}{2}$; output: $60^{\circ}$
Answer (1)
The sine function takes an angle as input and returns a ratio as output.
Evaluate sin ( 6 0 ∘ ) using the 30-60-90 triangle.
The value of sin ( 6 0 ∘ ) is 2 3 .
The correct input is 6 0 ∘ and the correct output is 2 3 .
Explanation
Understanding the Sine Function The problem asks us to identify the correct input and output for the sine function when the input angle is 6 0 ∘ . The sine function takes an angle as input (typically in degrees or radians) and returns a real number between -1 and 1 as output, representing the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.
Evaluating Sine of 60 Degrees We need to evaluate sin ( 6 0 ∘ ) . Recall the special right triangles, specifically the 30-60-90 triangle, where the sides are in the ratio 1 : 3 : 2 . In this triangle, the sine of the 6 0 ∘ angle is the ratio of the opposite side ( 3 ) to the hypotenuse (2). Therefore, sin ( 6 0 ∘ ) = 2 3 .
Determining the Correct Input and Output Based on the evaluation, when the input is 6 0 ∘ , the output is 2 3 .
Examples
In physics, when analyzing projectile motion, if a projectile is launched at an angle of 6 0 ∘ with an initial velocity, you need to calculate the vertical component of the velocity using sin ( 6 0 ∘ ) . Knowing that sin ( 6 0 ∘ ) = 2 3 allows you to determine the initial vertical velocity, which is crucial for calculating the projectile's range and maximum height. This principle applies in various fields, including sports (like calculating the launch angle for optimal performance in javelin throwing) and engineering (designing systems that involve angled launches or trajectories).
