Question

hence, the function ( k(\theta)=cos (2 \theta+3) ) has a period of ( pi ) radians.
find the periods of the following functions
a. ( g(x)=cos left(\frac{x}{4}
ight) )
the period of the function ( g ) is
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b. ( h(\theta)=sin (4(\theta+8)) )
the period of the function ( h ) is
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c. ( j(x)=sin left(\frac{2 x-3}{3}
ight) )
the period of the function ( j ) is
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question 6. points possible: 2
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Explanation:

Step1: Recall the formula for the period of a trigonometric function

The period of \(y = A\cos(Bx + C)\) or \(y=A\sin(Bx + C)\) is given by \(T=\frac{2\pi}{|B|}\).

Step2: Find the period of \(g(x)=\cos(\frac{x}{4})\)

For \(g(x)=\cos(\frac{x}{4})\), we have \(B = \frac{1}{4}\). Using the formula \(T=\frac{2\pi}{|B|}\), we substitute \(B=\frac{1}{4}\) into the formula:
\(T=\frac{2\pi}{\frac{1}{4}}=2\pi\times4 = 8\pi\).

Step3: Find the period of \(h(\theta)=\sin(4(\theta + 8))=\sin(4\theta+32)\)

For \(h(\theta)=\sin(4\theta + 32)\), we have \(B = 4\). Using the formula \(T=\frac{2\pi}{|B|}\), we substitute \(B = 4\) into the formula:
\(T=\frac{2\pi}{4}=\frac{\pi}{2}\).

Step4: Find the period of \(j(x)=\sin(\frac{2x-3}{3})=\sin(\frac{2}{3}x - 1)\)

For \(j(x)=\sin(\frac{2}{3}x-1)\), we have \(B=\frac{2}{3}\). Using the formula \(T=\frac{2\pi}{|B|}\), we substitute \(B=\frac{2}{3}\) into the formula:
\(T=\frac{2\pi}{\frac{2}{3}}=2\pi\times\frac{3}{2}=3\pi\).

Answer:

a. \(8\pi\)
b. \(\frac{\pi}{2}\)
c. \(3\pi\)