Questions: Determine the period of the following trigonometric function. y=-4 cos (8/5 x)-1

Solution Steps

The given trigonometric function is

\[ y = -4 \cos\left(\frac{8}{5} x\right) - 1 \]

In this function, we can identify the parameters as follows:

• Amplitude \( A = -4 \)
• Horizontal scaling factor \( B = \frac{8}{5} \)
• Horizontal shift \( C = 0 \)
• Vertical shift \( D = -1 \)

The period \( P \) of a cosine function is calculated using the formula:

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

Substituting the value of \( B \):

\[ P = \frac{2\pi}{\left|\frac{8}{5}\right|} = \frac{2\pi}{\frac{8}{5}} = \frac{2\pi \cdot 5}{8} = \frac{10\pi}{8} = \frac{5\pi}{4} \]

The period can also be expressed in terms of \( \pi \):

\[ P = 1.25\pi \]

Final Answer