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Chapter 5: Problem 71

Simplify the expression algebraically and use a graphing utility to confirm your answer graphically. $$\cos \left(\frac{3 \pi}{2}-x\right)$$

Short Answer

Expert verified
The final result would be, \(\cos(\frac{3 \pi}{2}-x) = \sin(x)\)

Step by step solution

01

Understand cosine properties

Recall the property of cosines. The cosine function has the property that \(\cos(-x) = \cos(x)\). This can be applied here because as the input \(-x\) changes to \(x\), the cosine function doesn鈥檛 change.
02

Change variables

Change variables so that the equation becomes \(\cos(\frac{3 \pi}{2} + (-x))\).
03

Apply cosine properties

Apply the cosine properties here, \(\cos(\frac{3 \pi}{2} + (-x))\) is equivalent to \(\cos((-\frac{3 \pi}{2} ) - x)\). Then we recall that \(\cos(-\frac{3 \pi}{2} - x)\) equals to \(\sin(x)\).
04

Verify graphically

To confirm the result graphically, plot y = cos(x) and y = sin(x) on the same set of axis. Notice how the graph of \(y = sin(x)\) is identical to the graph of \(y = cos(\frac{3 \pi}{2} - x)\). In particular, note how the graphs have the same shape and are shifted versions of one another.

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