Question

no calculator is allowed on this question.
i. the amplitude of $f(\theta)=\sin\theta$ is the distance from the origin to a maximum point of $f$.
ii. the amplitude of $f(\theta)=\sin\theta$ is the average of its maximum and minimum values.
iii. the amplitude of $f(\theta)=\sin\theta$ is the distance between the horizontal axis and a minimum point on $f$.
identify the true statement(s) about the amplitude of $f(\theta)=\sin\theta$.
select one answer
a i only
b ii only
c iii only
d i, ii, and iii

Explanation:

1. Recall the definition of the amplitude of a sinusoidal function \( y = A\sin(\theta) \). The amplitude is the absolute value of \( A \), which represents the maximum distance from the midline (horizontal axis) to a maximum or minimum point of the function.
2. For \( f(\theta)=\sin\theta \), the maximum value is \( 1 \) and the minimum value is \( - 1 \), and the midline is \( y = 0 \) (the horizontal axis).

• Statement I: The maximum point of \( f(\theta)=\sin\theta \) is at \( y = 1 \), and the distance from the origin (on the horizontal axis, \( y = 0 \)) to this maximum point is \( |1 - 0|=1 \), which is the amplitude. So statement I is true.
• Statement II: The maximum value of \( f(\theta)=\sin\theta \) is \( 1 \) and the minimum value is \( - 1 \). The average of the maximum and minimum values is \( \frac{1+( - 1)}{2}=\frac{0}{2} = 0 \)? Wait, no, wait. Wait, actually, the amplitude can also be calculated as \( \frac{\text{max value}-\text{min value}}{2} \). For \( \sin\theta \), max is \( 1 \), min is \( - 1 \), so \( \frac{1-( - 1)}{2}=\frac{2}{2}=1 \), which is the amplitude. So the average of the maximum and minimum values (in the sense of \( \frac{\text{max}-\text{min}}{2} \)) gives the amplitude. So statement II is true.
• Statement III: The minimum point of \( f(\theta)=\sin\theta \) is at \( y=-1 \), and the distance from the horizontal axis (\( y = 0 \)) to this minimum point is \( |-1 - 0| = 1 \), which is the amplitude. So statement III is true.

Since all three statements I, II, and III are true, the correct answer is D.

Answer:

D. I, II, and III