
Find the period of
Answer
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Hint: The length of one complete cycle is called a period of a trigonometric function. The cosine function is even and the sine function is odd. A function is even, if A function is odd, if
Complete step-by-step solution:
We know that the period of a trigonometric function is the length of one complete cycle.
Consider the given trigonometric function
Now, let us suppose that
Also, we know that the cosine function is even and the sine function is odd.
We have already learnt that, for an even function
Similarly, for an odd function
So, we will get and
Now we use these identities to find the period of the given function
Also, we know that
Similarly,
Because, in the second quadrant, the sine function is positive and the cosine function is negative.
Since we have these identities, we can find that
From the above, we will get the following
Now we can use the properties of even functions to get,
Because, we will get and
And now we can see that
And this will give us the following
Thus, we will get
Hence, we can conclude that the period of the given function is
Note: The period of the cosine function is That is, the period of The period of the sine function is also That is, the period of The period can be found as follows:
Period of and
Now the period of
That is, Since
Complete step-by-step solution:
We know that the period of a trigonometric function is the length of one complete cycle.
Consider the given trigonometric function
Now, let us suppose that
Also, we know that the cosine function is even and the sine function is odd.
We have already learnt that, for an even function
Similarly, for an odd function
So, we will get
Now we use these identities to find the period of the given function
Also, we know that
Similarly,
Because, in the second quadrant, the sine function is positive and the cosine function is negative.
Since we have these identities, we can find that
From the above, we will get the following
Now we can use the properties of even functions to get,
Because, we will get
And now we can see that
And this will give us the following
Thus, we will get
Hence, we can conclude that the period of the given function
Note: The period of the cosine function is
Period of
Now the period of
That is,
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