Chapter 13.5, Problem 45PPSE

Solution

i.

To graph: The given functions in the same viewing window and determine the translation required to transform a given function into a specific function.

Translation: $\frac{\pi }{2}$ unit right

Given:

The functions $y=\cos \theta$ and $y=\cos \left(\theta -\frac{\pi }{2}\right)$ .

Calculation:

The graph of the both the function is given below in the same viewing window:

  

Here in the graph, the blue line represents the function $y=\cos \theta$ and the yellow line represent the function $y=\cos \left(\theta -\frac{\pi }{2}\right)$ .

From the graph, it is understood that by translating $y=\cos \theta$ to $\frac{\pi }{2}$ unit right the function $y=\cos \left(\theta -\frac{\pi }{2}\right)$ can be obtained.

Conclusion:

Hence, the graph is given below and the required translation is $\frac{\pi }{2}$ unit right.

ii.

To graph: The given function in the same viewing window.

Both the functions are equivalent.

Given:

The functions $y=\cos \left(\theta -\frac{\pi }{2}\right)$ and $y=\sin \theta$ .

Calculation:

The graph of both the functions are given below in same viewing window:

  

The curve of both the functions are overlaps each other and only one curve is showing. So, it can be concluded that both the functions are equivalent.

Conclusion:

Hence, the graph of both the functions are given above and both the functions are equivalent.

iii.

To find: how to write sine function as cosine function.

  $y=\cos \left(\frac{\pi }{2}-x\right)$

Given:

The sine function $y=\sin \theta$ .

Calculation:

If the curve of cosine function translated $\frac{\pi }{2}$ unit right then it is equivalent to the sine function. So, sine function can be represented as cosine by writing the function as follows.

  $y=\cos \left(\frac{\pi }{2}-x\right)$

Conclusion:

Hence, the cosine function $y=\cos \left(\frac{\pi }{2}-x\right)$ is equivalent to the sine function.