Question

Determine the period of the following trigonometric function.
y, equals, 3, cosine, left parenthesis, start fraction, 8, x, divided by, 7, end fraction, plus, 3, right parenthesis
y=3cos(
7
8x
​
+3)

Answer 1

To determine the period of the trigonometric function given by

\[
y = 3 \cos\left(\frac{8}{7} x + 3\right),
\]

we need to identify the coefficient of \(x\) inside the cosine function.

The general form of the cosine function is:

\[
y = A \cos(Bx + C),
\]

where:
- \(A\) is the amplitude,
- \(B\) is the coefficient that affects the period of the function,
- \(C\) is the phase shift.

The period \(P\) of the cosine function is given by:

\[
P = \frac{2\pi}{|B|}.
\]

In our case, we have:

\[
B = \frac{8}{7}.
\]

Now, we can calculate the period:

\[
P = \frac{2\pi}{\left|\frac{8}{7}\right|} = \frac{2\pi}{\frac{8}{7}} = 2\pi \cdot \frac{7}{8} = \frac{14\pi}{8} = \frac{7\pi}{4}.
\]

Thus, the period of the function \(y = 3 \cos\left(\frac{8}{7} x + 3\right)\) is

\[
\boxed{\frac{7\pi}{4}}.
\]