Problem 95 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Step 5: 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 \(\frac{\pi}{2}\).