QuestionDetermine the period of the function $y=2 \cos 2x + \sin x$ .

Studdy Solution

STEP 1

Assumptions1. The function is y= \cosx + \sin x . The period of a function is the distance over which the function's graph completes one full cycle3. The period of a basic sine or cosine function is $\pi$
4. The period of a function $y=a \sin bx$ or $y=a \cos bx$ is $\frac{\pi}{|b|}$

STEP 2

The given function is a sum of two trigonometric functions, each with a different coefficient for $x$ . Let's find the period of each part of the function separately.
The first part of the function is $2 \cos2x$ . The period of this function can be found by using the formula $\frac{2\pi}{|b|}$ , where $b$ is the coefficient of $x$ . $Period1 = \frac{2\pi}{|2|}$

STEP 3

Calculate the period of the first part of the function.
$Period1 = \frac{2\pi}{2} = \pi$

STEP 4

The second part of the function is $\sin x$ . The period of this function can be found by using the formula $\frac{2\pi}{|b|}$ , where $b$ is the coefficient of $x$ . $Period2 = \frac{2\pi}{|1|}$

STEP 5

Calculate the period of the second part of the function.
$Period2 = \frac{2\pi}{1} =2\pi$

STEP 6

The period of the sum of two functions is the least common multiple (LCM) of the periods of the individual functions. So, we need to find the LCM of $\pi$ and $2\pi$ .
Since $2\pi$ is a multiple of $\pi$ , the LCM of $\pi$ and $2\pi$ is $2\pi$ .
Therefore, the period of the function $y=2 \cos2x + \sin x$ is $2\pi$ .

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