Chapter 12.3, Problem 47HP

To determine

To find: Can the value of $\theta$ be ${225}^0$ .

Answer to Problem 47HP

  $\theta$ cannot be ${225}^0$

Explanation of Solution

Given information:

  $\sin \left(\theta \right)=\frac{\sqrt{2}}{2}$ and $\tan \left(\theta \right)=-1$

Concept:

If $\theta$ is the non-quadrantal angle in standard position, its reference angle ${\theta }^{\text{'}}$ is the acute angle formed by the terminal side of $\theta$ and the x-axis.

The ruled for finding the measures of reference angles for $0^0<\theta <{360}^0$ or $0^0<\theta <2\pi$ are shown

For Quadrant − I

  ${\theta }^{\text{'}}=\theta$

For Quadrant − II

  ${\theta }^{\text{'}}={180}^0-\theta$ or ${\theta }^{\text{'}}=\pi -\theta$

For Quadrant − III

  ${\theta }^{\text{'}}=\theta -{180}^0$ or ${\theta }^{\text{'}}=\theta -\pi$

For Quadrant − IV

  ${\theta }^{\text{'}}={360}^0-\theta$ or ${\theta }^{\text{'}}=2\pi -\theta$

Given

  $\sin \left(\theta \right)=\frac{\sqrt{2}}{2}$ and $\tan \left(\theta \right)=-1$

Here from the given information we can say that

Value of sine function is positive and Value of tan function is negative.

This will be possible only if the value of $\theta$ lies in Quadrant-II.

As given $\theta ={225}^0$

So, it means ${225}^0$ lies in Quadrant-III

To get Value of sine function is positive and Value of tan function is negative.

  $\theta$ must be in Quadrant-II.

So, $\theta$ cannot be ${225}^0$ .