91Ó°ÊÓ

Amplitude is a key concept when studying trigonometric functions like sine and cosine. It refers to the height of the wave, specifically how far the function's value travels above and below its average value. In simpler terms, amplitude is the maximum distance from the horizontal axis to the peak or trough of the wave.

In the equation given, which is in the form of \( y = a\cos(bx + c) \), the amplitude \( a \) is represented by the coefficient in front of the cosine function. For the equation \( y = \cos(x + \frac{\pi}{2}) \), the coefficient is 1. Therefore, the amplitude is absolute value of 1, which is:

• Amplitude = |1| = 1

This means the wave reaches 1 unit above and 1 unit below the horizontal axis. Remember, the amplitude tells us nothing about the position along the x-axis or the rate at which the wave moves. It's purely about vertical stretch or compression.

The period of a trigonometric function describes how long it takes for the function to repeat its cycle. For the cosine function, the period reflects the distance along the x-axis before the function starts to repeat. The period formula for the function \( y = a\cos(bx + c) \) is \( \frac{2\pi}{b} \).

In our equation, \( y = \cos(x + \frac{\pi}{2}) \), the coefficient \( b \) of \( x \) is 1. This makes the calculation straightforward:

• Period = \( \frac{2\pi}{1} = 2\pi \)

This period means that after every \( 2\pi \) units along the x-axis, the same wave pattern repeats. It's like drawing the wave, knowing it will look the exact same every \( 2\pi \) interval. Understanding the period helps us know how "stretched out" a cycle is compared to the basic cosine function.

Phase shift involves figuring out where the wave starts along the x-axis. It refers to a horizontal shift that causes the wave to start at a different point than usual. Phase shift is rooted in the "inside" value added or subtracted in the function, represented as \( c \) in \( y = a\cos(bx + c) \).

To determine the phase shift, use the formula \( -\frac{c}{b} \). For \( y = \cos(x + \frac{\pi}{2}) \), \( c \) is \( \frac{\pi}{2} \) and \( b \) is 1, leading to:

• Phase Shift = \(-\frac{\frac{\pi}{2}}{1} = -\frac{\pi}{2}\)

The negative sign reveals the wave shifts to the left by \( \frac{\pi}{2} \) units. Essentially, understanding phase shift helps us reposition the starting point of the wave, adjusting where it begins its cycle along the x-axis. This alteration aligns the graph with any phase changes, crucial for accurately sketching trigonometric graphs.