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Chapter 5: Problem 95

Describe a relationship between the graphs of \(y=\sin x\) and \(y=\cos x\)

Short Answer

Expert verified
The trigonometric functions \(\sin x\) and \(\cos x\) are identical in shape, having the same oscillation patterns, periods and frequencies. Their graphs however differ by a phase shift of \(\frac{\pi}{2}\) units, with \(\cos x\) being a shift of \(\sin x\) to the left by \(\frac{\pi}{2}\). This is the relationship between the graphs of \(y=\sin x\) and \(y=\cos x\).

Step by step solution

01

Plotting the Graphs

Plot the graphs of the functions \(y=\sin x\) and \(y=\cos x\) in the same Cartesian plane. These trigonometric functions have similar wave-like patterns or oscillations.
02

Comparing the Shapes

Observe both the plots. Both \(\sin x\) and \(\cos x\) have equivalent shapes. They are periodic and continuously oscillate between -1 and 1.
03

Comparing Period and Frequency

Note that both \(\sin x\) and \(\cos x\) have the same period and frequency. Their periods and frequencies are \(2\pi\) and \(\frac{1}{2\pi}\) respectively.
04

Analysing Phase Shift

Observe the phase shift between the two functions. The \(\cos x\) graph is a horizontal shift of the \(\sin x\) graph to the left by \(\frac{\pi}{2}\) units. In other terms, \(\cos x\) is the same as \(\sin (x + \frac{\pi}{2})\). This is known as a phase shift.
05

Concluding the Relationship

To conclude, \(\sin x\) and \(\cos x\) are identical in shape, period and frequency but they differ by a phase shift of \(\frac{\pi}{2}\).

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