Problem 42 - Algebra and Trigonometry

Question

\(29-42\) . Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=\cos \left(\frac{\pi}{2}-x\right) $$

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(2\pi\), Phase Shift: 0. Graph: Sine wave from \(0\) to \(2\pi\).

Step 1: Identify the Trigonometric Function

The given function is \( y = \cos\left(\frac{\pi}{2} - x\right) \). This is a cosine function.

Step 2: Use Trigonometric Identity

We know the identity \( \cos(\frac{\pi}{2} - x) = \sin(x) \). Therefore, the function simplifies to \( y = \sin(x) \).

Step 3: Determine the Amplitude

The standard function \( y = \sin(x) \) has an amplitude of 1. Therefore, the amplitude of the given function is 1.

Step 4: Determine the Period

The standard period of \( \sin(x) \) is \( 2\pi \). Since there is no coefficient affecting the \( x \) in \( \sin(x) \), the period remains \( 2\pi \).

Step 5: Determine the Phase Shift

The phase shift is determined by any changes inside the function’s argument relative to a standard sine function. Since our function simplifies directly to \( \sin(x) \), the phase shift is 0.

Step 6: Sketch One Complete Period

One complete period of \( y = \sin(x) \) starts at (0,0), peaks at \( (\frac{\pi}{2}, 1) \), returns to zero at \( \pi \), hits its lowest point at \( (\frac{3\pi}{2}, -1) \), and completes the period at \( 2\pi \). The graph is a wave completing one cycle over \( 0 \) to \( 2\pi \).

Key Concepts

Amplitude

In trigonometric functions, the amplitude represents the peak value of the function's wave from its average level. For the given function, which simplifies to \( y = \sin(x) \), the amplitude is quite straightforward to determine. The amplitude is found by examining the coefficient in front of the sine function. Amplitude tells us how "tall" (or "deep") the wave of the function gets from its midline, which is the horizontal line (usually the x-axis) around which the wave oscillates.

• The amplitude is the absolute value of the coefficient of the trigonometric function.
• In the case of \( y = \sin(x) \), there is no number multiplying the sine function, meaning the coefficient is 1.
• Hence, the amplitude is \(1\), indicating that the peak of the wave reaches 1 unit above and below the center line of the wave.

Knowing the amplitude is crucial as it informs us of the maximum displacement from the center line that the wave reaches which, in practical situations, could translate to sound intensity, light brightness, etc.

Period

The period of a trigonometric function defines how long it takes for the function to complete one full cycle. It helps us understand the regularity at which the wave repeats its shape. Typically, to determine the period of a sine function, we utilize the formula \( \text{Period} = \frac{2\pi}{b} \) where \( b \) is the coefficient multiplied by \( x \) in the trigonometric function.Here, the function \( y = \sin(x) \) is in its standard form.

• There is no coefficient in front of \( x \), which means \( b = 1 \).
• Thus, the period is \(\frac{2\pi}{1} = 2\pi\).

This suggests that one full wave or cycle spans from \( 0 \) to \( 2\pi \), completing a pattern that consists of rising from the center line, peaking, returning to center, dipping to a trough, and finally returning to the center line.

Phase Shift

Phase shift is a measure of how much the graph of the function is moved horizontally from its standard position. In other words, it tells us where the cycle of the function starts along the x-axis in comparison to the normal start point. For the function \( y = \sin(x) \), no alterations inside the function's argument indicate no horizontal movement.

• The general form \( y = a\sin(bx - c) + d \) provides a phase shift calculated by \( \frac{c}{b} \).
• Since \( c = 0 \) in our function \( y = \sin(x) \), there is \(0\) phase shift.

Understanding phase shift is crucial in scenarios where the timing or starting point of a wave-like occurrence, such as sound waves or ocean waves, is necessary to interpret or predict correctly.