Problem 37 - Algebra 2

Question

a. Graph \(y=\cos \theta\) and \(y=\cos \left(\theta-\frac{\pi}{2}\right)\) in the interval from 0 to 2\(\pi .\) What translation of the graph of \(y=\cos \theta\) produces the graph of \(y=\cos \left(\theta-\frac{\pi}{2}\right) ?\) b. Graph \(y=\cos \left(\theta-\frac{\pi}{2}\right)\) and \(y=\sin \theta\) in the interval from 0 to 2\(\pi .\) What do you notice? c. Explain how you could rewrite a sine function as a cosine function.

Step-by-Step Solution

Answer

a) The graph of \(y=\cos \left(θ-\frac{\pi }{2}\right)\) is a translation of the graph of \(y=\cos θ \) π/2 units to the right. b) The graphs of \(y= \cos\left(θ -\frac{\pi }{2}\right)\) and \(y=\sin θ \) are identical. c) A sine function can be rewritten as a cosine function by shifting the graph π/2 units to the right - so \(\sin θ \) can be written as \(\cos\left(θ -\frac{\pi }{2}\right) \).

Step 1 - Graph y=cos(θ) and y=cos(θ-π/2)

Plot the two functions \(y=\cos (θ) \) and \(y=\cos (θ-\frac{\pi }{2}) \) within the interval from 0 to 2π. Here, note that \(y=\cos (θ-\frac{\pi }{2}) \) is just the \(y=\cos θ \) graph, but shifted π/2 units to the right.

Step 2 - Graph y= cos(θ-π/2) and y=sin(θ)

Again, plot the two functions \(y=\cos (θ-\frac{\pi }{2}) \) and \(y=\sin θ \) within the interval from 0 to 2π. By comparing both graphs, we can see that they are identical.

Step 3 - Relationship between sine and cosine functions

Observe that a sine function can be transformed into a cosine function by shifting the graph π/2 to the left or right. Therefore, we can rewrite \(\sin θ \) as \(\cos\left(θ -\frac{\pi }{2}\right) \).

Key Concepts

Trigonometric Function Graphs

Trigonometric functions, especially sine and cosine, play a fundamental role in mathematics, primarily in trigonometry. These functions represent periodic phenomena and are pivotal in describing waves, circular motions, and oscillations.
The graph of the cosine function, denoted as \(y=\cos \theta\), is a wave that repeats every \(2\pi\) units. This is referred to as its period. For the basic cosine function, the curve starts at its maximum of 1 when \(\theta = 0\), dips to -1, and returns to 1 over this interval. It's a smooth, continuous wave often called the "cosine wave".

• The key points to identify on the cosine graph are at \(0, \pi, \text{ and } 2\pi\), where it reaches peaks and valleys.
• The amplitude, the height from the midline to the peak, of the basic cosine and sine graphs is always 1.
• The midpoint, or the line that the graph oscillates around, is the x-axis for the function \(y=\cos \theta\).

Recognizing these characteristics in graphing can simplify dealing with transformations and translations of trigonometric graphs.

Translation of Trigonometric Graphs

Graph translation involves shifting a graph horizontally or vertically. For trigonometric functions, horizontal translations or "phase shifts" are especially noteworthy.
When graphing \(y=\cos(\theta-\frac{\pi}{2})\), the cosine function \(y=\cos \theta\) is shifted to the right by \(\frac{\pi}{2}\) units. This means every point on the graph of the original cosine function moves \(\frac{\pi}{2}\) to the right.

• A positive angle subtraction inside the cosine function denotes a shift to the right, while addition indicates a leftward shift.
• This shift changes the phase of the waveform without altering its shape or amplitude.

Understanding translations like these are crucial when dealing with problems involving time shifts or phase differences in fields such as physics and engineering.

Relationship Between Sine and Cosine

The sine and cosine functions are closely related. Essentially, the sine function is a horizontal shift of the cosine function, and vice versa.
Observing the graphs of \(y=\sin \theta\) and \(y=\cos(\theta-\frac{\pi}{2})\), it becomes clear that they are identical. This identical behavior arises because shifting the cosine curve to the right by \(\frac{\pi}{2}\) yields the sine curve.

• Mathematically, this relationship can be expressed as \(y=\sin \theta = \cos(\theta - \frac{\pi}{2})\).
• This transformation highlights the fact that both functions are essentially the same wave, represented differently based on their phase shift.
• For any angle \(\theta\), the sine value equals the cosine of \(\theta - \frac{\pi}{2}\).

This fundamental transformation is key to converting between sine and cosine functions in trigonometric identities and equations.

Cosine Function Transformations

Transformations in trigonometry involve changing a function's graph through translations, dilations, and reflections.
The exercise shows that \(y=\cos(\theta-\frac{\pi}{2})\) is a transformed version of \(y=\cos \theta\) through horizontal translation. Transformations don't always involve only shifts; they can also include scaling and mirroring.

• To scale the cosine graph, we multiply \(\theta\) by a coefficient, changing the period of the wave. For example, \(y=\cos(2\theta)\) halves the period.
• Reflections occur when the function's sign is changed, such as \(y=-\cos \theta\), flipping the graph over the x-axis.
• Amplitude changes involve multiplying the entire function by a constant, such as \(y=2\cos \theta\), which doubles the wave's height.

Understanding these transformations allows one to manipulate and graph cosine functions for diverse applications in computer graphics, signal processing, and more.