Chapter 5.2, Problem 96E

To determine

To explain: Why the given equation is not an identity and find one value of the variable for which the equation is not true.

Explanation of Solution

Given information:

The following equation:

  $\sin \theta =\sqrt{1-{\cos }^2\theta }$

Formula used:

The following identity:

  ${\text{sin}}^2\theta +{\cos }^2\theta =1$

The equation can be simplified as:

  $\begin{array}{l}\sin \theta =\sqrt{1-{\cos }^2\theta }\text{ [Given]}\\ \text{ =}\sqrt{{\sin }^2\theta }\text{ [Pythagorean identity]}\\ \ne \pm \sin \theta \text{ [Square root]}\end{array}$

The left hand side is not equal to the right hand side. Hence the equation is not an identity.

Let $\theta =\frac{3\pi }{2}$

Substituting value of $\theta$ on left hand side of the equation,

  $\begin{array}{l}\sin \theta \\ =\sin \frac{3\pi }{2}\text{ [Substitute value of }\theta \text{]}\\ =-1\text{ [Put value of }\sin \frac{3\pi }{2}]\end{array}$

Substituting value of $\theta$ on right hand side of the equation,

  $\begin{array}{l}\sqrt{1-{\cos }^2\theta }\\ =\sqrt{1-{\cos }^2\frac{3\pi }{2}}\text{ [Substitute value of }\theta \text{]}\\ =\sqrt{1-0}\text{ [Put value of }{\cos }^2\frac{3\pi }{2}]\\ \text{=}\pm \text{1 [Square root] }\end{array}$

The value on the left hand side is not equal to the value on the right hand side.

Hence for $\theta =\frac{3\pi }{2}$ , the equation is not true

Also, the given equation is not an identity because the expressions on the left hand side and the right hand side when plotted on a graph, do not coincide.

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions ${\text{Y}}_1=\sin \theta$ and ${\text{Y}}_2=\sqrt{1-{\cos }^2\theta }$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

The graphs of the left hand side expression and the right hand side expression do no coincide.