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Period of sin^2x +cos^2x?
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Period of sin^2x +cos^2x?
Period of sin^2x cos^2x
Introduction
When we talk about period of a function, we are referring to the length of the interval over which the function repeats itself. In this case, we are interested in finding the period of the function sin^2x cos^2x.
Method
To find the period of sin^2x cos^2x, we can use the following steps:
- Use the trigonometric identity sin^2x + cos^2x = 1 to simplify the function:
- sin^2x cos^2x = (sin^2x)(1 - sin^2x) = sin^2x - sin^4x
- Notice that sin^2x - sin^4x is an even function, which means that it is symmetric about the y-axis. This means that we only need to find the period of sin^2x - sin^4x for one half of the interval (either the positive or negative values of x) and then double it to get the full period.
- Consider the function f(x) = sin^2x - sin^4x. To find the period of f(x), we need to find the smallest positive value of p for which f(x + p) = f(x) for all x.
- Let's begin by finding f(x + p) for some arbitrary value of p:
- f(x + p) = sin^2(x + p) - sin^4(x + p)
- = sin^2x cos^2p + cos^2x sin^2p - (sin^2x cos^2p + cos^2x sin^2p)^2
- = sin^2x cos^2p + cos^2x sin^2p - sin^4x cos^4p - 2sin^2x cos^2p sin^2p - 2cos^2x sin^2p cos^2p
- = sin^2x cos^2p + cos^2x sin^2p - sin^4x cos^4p - 2sin^2x cos^2x sin^2p cos^2p
- Now we can set f(x + p) equal to f(x) and simplify:
- sin^2x - sin^4x = sin^2x cos^2p + cos^2x sin^2p - sin^4x cos^4p - 2sin^2x cos^2x sin^2p cos^2p
- sin^2x (1 - cos^2p - cos^2x sin^2p) = sin^4x (1 - cos^4p - 2cos^2x sin^2p cos^2p)
- sin^2x (sin^2p + cos^2p - cos^2p
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Period of sin^2x +cos^2x?
Question Description
Period of sin^2x +cos^2x? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Period of sin^2x +cos^2x? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Period of sin^2x +cos^2x?.
Period of sin^2x +cos^2x? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Period of sin^2x +cos^2x? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Period of sin^2x +cos^2x?.
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