Question
What is the period of the graph of \( y=\sin x \) ? a \( \frac{\pi}{2} \) b \( \frac{\pi}{4} \) c \( \pi \) d \( 2 \pi \)
Ask by Fernandez French. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The period of the graph of \( y = \sin x \) is \( 2\pi \).
Solution
To determine the period of the function \( y = \sin x \), we can use the definition of the period of a sine function. The period \( T \) of a sine function is the length of one complete cycle of the wave.
The general form of the sine function is given by:
\[
y = A \sin(Bx + C) + D
\]
where:
- \( A \) is the amplitude,
- \( B \) affects the period,
- \( C \) is the phase shift,
- \( D \) is the vertical shift.
The period \( T \) can be calculated using the formula:
\[
T = \frac{2\pi}{|B|}
\]
In the case of \( y = \sin x \), we can see that \( B = 1 \). Therefore, the period is:
\[
T = \frac{2\pi}{|1|} = 2\pi
\]
Thus, the period of the graph of \( y = \sin x \) is \( 2\pi \).
The correct answer is:
d \( 2 \pi \)
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Simplify this solution Bonus Knowledge
The period of the graph of \( y = \sin x \) is \( 2\pi \). This means that the sine function repeats its values every \( 2\pi \) units along the x-axis. So if you start at any point on the graph, after moving \( 2\pi \) to the right, you'll land on the exact same point on the curve. In the world of trigonometric functions, understanding the period is key! For example, if you're modeling sound waves or vibrations, knowing that \( \sin x \) repeats every \( 2\pi \) helps in predicting and analyzing patterns, making it extremely useful in fields like physics and engineering.
