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Magazine Chapter 6: Problem 71
What is the period of the function \(\sin ^{2} x ?\)
Short Answer
Expert verified
The period of the function \( \sin^2 x \) is \( \pi \).
Step by step solution
01
Period of the sine function
The basic sine function, \( \sin x \), has a period of \( 2\pi \). This means that for any value of \(x\),
\[ \sin(x + 2\pi) = \sin(x) .\]
02
Analyzing the square of the sine function
Now, we need to consider the given function: \( \sin^2 x = (\sin x)^2 \). We want to find the smallest value for which the function repeats its values, i.e., we are looking for the value \(p\) such that
\[ \sin^2(x + p) = \sin^2(x) .\]
03
Use properties of the sine function
We know that the sine function is an odd function, meaning
\[ \sin(-x) = -\sin(x) .\]
This means that if \(\sin(x) = \sin(-x)\), then \(\sin^2(x)=\sin^2(-x)\), and these equalities hold for any value of \(x\).
04
Determine the period of the square of the sine function
Using our knowledge of the sine function, we can find p by examining the sine function's properties. As mentioned before,
\[ \sin^2(x) = \sin^2(-x) .\]
Therefore, if we substitute \( -x \) for the \( x \) in the sine function's period definition, we get
\[ \sin(x+\pi) = \sin(-x). \]
Now, squaring both sides, we have
\[ \sin^2(x+\pi) = \sin^2(-x). \]
Comparing this equation with \( \sin^2(x+p) = \sin^2(x) \), we find that \(p = \pi\).
05
Write the final answer
So, the period of the function \( \sin^2 x \) is \( \pi \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
trigonometric functions
Trigonometric functions play a crucial role in mathematics, especially in studying periodic phenomena. They include functions like sine, cosine, and tangent, each offering a unique relationship between angles and ratios of triangle sides. One of the fundamental properties of trigonometric functions is periodicity, meaning they repeat their values in regular intervals. For instance:
- The sine function, \( \sin x \), has a period of \( 2\pi \), interpreting that its graph repeats every \( 2\pi \) radians.
- The cosine function, \( \cos x \), shares this same period with sine.
sine function
The sine function, represented as \( \sin x \), is one of the most commonly used trigonometric functions. It maps the angle \( x \) to a value between -1 and 1. The function is defined on the entire real line and is periodic with a fundamental period of \( 2\pi \). This means:
- If you add \( 2\pi \) to any angle \( x \), the sine value remains unchanged: \( \sin(x + 2\pi) = \sin(x) \).
- This property helps predict the behavior of \( \sin x \) over an infinitely long interval.
function transformations
Function transformations can modify a function's graph to stretch, shift, reflect, or compress it. For the sine function, transformations are particularly important:
- Stretching or compressing the graph vertically by multiplying \( \sin x \) by a constant.
- Shifting the graph horizontally or vertically through addition or subtraction.
- Reflecting over the x-axis or y-axis by multiplying the function by -1.
mathematical properties
Mathematical properties are rules and features that describe how mathematical concepts behave. In trigonometry, the sine function has several important properties:
- Odd nature: \( \sin(-x) = -\sin(x) \), showcasing symmetry about the origin.
- Range: The sine values lie between -1 and 1 for all \( x \).
- Periodicity: Repeats every \( 2\pi \) in the case of \( \sin x \) and every \( \pi \) for \( \sin^2 x \).
