Problem 47 - Algebra 2

Question

State the amplitude, period, and phase shift of each function. Then graph each function. $$ y=\sin \left(\theta-45^{\circ}\right) $$

Step-by-Step Solution

Answer

Amplitude: 1, Period: 360°, Phase Shift: 45° to the right.

Step 1: Determine the Amplitude

The amplitude of the function \[ y = \sin(\theta-45^{\circ}) \] is determined by the coefficient in front of the \( \sin \) function. In this case, there is no coefficient other than 1, so the amplitude is 1.

Step 2: Determine the Period

The period of a sine function \( y = \sin(b\theta) \) is given by \( \frac{360^{\circ}}{b} \). Here, \( b = 1 \), so the period is \( \frac{360^{\circ}}{1} = 360^{\circ} \).

Step 3: Determine the Phase Shift

The phase shift of a sine function of the form \( y = \sin(\theta - \Phi) \) is \( \Phi \). In this function, \( \Phi = 45^{\circ} \), so the phase shift is \( 45^{\circ} \) to the right.

Step 4: Graph the Function

To graph \( y = \sin(\theta-45^{\circ}) \), start by plotting the basic sine wave with an amplitude of 1 and a period of \( 360^{\circ} \). Then shift the entire graph \( 45^{\circ} \) to the right to account for the phase shift. The points of interest like peaks and zero crossings will occur at \( 45^{\circ} \) intervals compared to the standard sine wave.

Key Concepts

Sine Function

The sine function is one of the most fundamental trigonometric functions, typically represented as \( y = \", \sin\theta\", \). It's visualized as a smooth, continuous wave. The sine function oscillates between -1 and 1, and it is widely used in mathematics, physics, and engineering due to its periodic nature.

• Wave Nature: The sine wave looks like smooth, rolling hills that repeat after a regular interval.
• Range: The output or range of the sine function is limited to values between -1 and 1.
• Trigonometric Use: In right-angled triangles, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse.
• Applications: Besides triangles, sine functions model various phenomena such as sound waves, light waves, and tides.

Amplitude

Amplitude in the context of the sine function refers to the height of the wave from the middle (equilibrium position) to its peak. For a general sine function, \( y = A \sin(b\theta + c) \), the amplitude is given by the value of \( A \). This amplitude determines how "tall" or "short" the wave appears.

• Standard Amplitude: If no coefficient is present, as in \( y = \sin\theta \), the amplitude is 1.
• Effect: Amplitude affects the strength or intensity of the wave. More amplitude means higher peaks and deeper troughs.
• Zero Amplitude: If the amplitude is zero, the wave would essentially be a flat line, showing no variation.

Being able to recognize the amplitude helps in graphing the wave accurately and understanding its physical significance in practical applications.

Period

The period of a sine function describes how long it takes for the wave to complete one full cycle before repeating. In the graph of the sine wave, the period is represented as the length between two consecutive peaks or troughs. Mathematically, for \( y = \sin(b\theta) \), the period is calculated as \( \frac{360^{\circ}}{b} \).

• Standard Period: The basic sine function \( y = \sin\theta \) has a period of \( 360^{\circ} \), meaning it repeats itself every \( 360^{\circ} \).
• Adjusting the Period: Changes in the value of \( b \) compress or stretch the wave horizontally. A higher value of \( b \) leads to a shorter period, producing more frequency.
• Real-Life Examples: Understanding the period helps in areas like signal processing, where matching the period to certain frequencies is crucial.

Phase Shift

The phase shift determines where the wave starts along the horizontal axis. It is what shifts the standard position of the sine wave to the left or right. In the function \( y = \sin(b\theta - \Phi) \), the phase shift is \( \Phi \).

• Reading the Shift: A positive \( \Phi \) would indicate a rightward shift, whereas a negative \( \Phi \) shifts the wave to the left.
• Influence of Phase Shift: It affects the initial position of the cycle. Points where the sine wave crosses the axis, reaches peaks, or bottoms out are all moved horizontally.
• Graphical Interpretation: When graphing, phase shift adjustments help align sine waves to match periodic cycles in real data like daylight patterns or seasonal changes.

In our exercise, the phase shift is \( 45^{\circ} \) to the right. Knowing this allows you to accurately graph the function by starting each cycle \( 45^{\circ} \) later than the usual standard.