Determine the amplitude, period, and phase shift of the following trigonometric equation. \[ y=\frac{1}{3} \cos (x+5) \] Answer

To analyze the trigonometric equation \[ y = \frac{1}{3} \cos (x + 5), \] we can identify the amplitude, period, and phase shift by comparing it to the standard form of a cosine function: \[ y = A \cos(B(x - C)) + D, \] where: - \( A \) is the amplitude, - \( \frac{2\pi}{B} \) is the period, - \( C \) is the phase shift, - \( D \) is the vertical shift. ### Step 1: Amplitude The amplitude \( A \) is the absolute value of the coefficient in front of the cosine function. Here, we have: \[ A = \left| \frac{1}{3} \right| = \frac{1}{3}. \] ### Step 2: Period The period of the cosine function is given by the formula \( \frac{2\pi}{B} \). In our equation, the coefficient \( B \) is 1 (since there is no coefficient in front of \( x \)). Therefore, the period is: \[ \text{Period} = \frac{2\pi}{1} = 2\pi. \] ### Step 3: Phase Shift The phase shift is determined by the term inside the cosine function. The standard form has \( (x - C) \), so we need to rewrite \( (x + 5) \) as \( (x - (-5)) \). Thus, we have: \[ C = -5. \] The phase shift is given by \( C \), which means the phase shift is: \[ \text{Phase Shift} = -5 \text{ (to the left)}. \] ### Summary Putting it all together, we have: - Amplitude: \( \frac{1}{3} \) - Period: \( 2\pi \) - Phase Shift: \( -5 \) (to the left) Thus, the final answer is: - Amplitude: \( \frac{1}{3} \) - Period: \( 2\pi \) - Phase Shift: \( -5 \) (to the left)