Chapter 4, Problem 14RE

Solution

To find: The smallest positive angle measured by the terminal side of the point $\left(-3,-3\right)$ in both radians and degrees.

The smallest positive angle made by the terminal side of the point $\left(-3,-3\right)$ is $225°$ in degrees and $\frac{5\pi }{4}$ in radians.

Given information: The given point is $\left(-3,-3\right)$ .

Formula used: Consider $\theta$ as the angle at the standard position, $P\left(x,y\right)$ as the point at the terminal side of the angle, and $r$ as the distance from the origin to the point P making the angle $\theta$ with the $x$ -axis.

  $\tan \theta =\frac{y}{x}$

Calculation:

Compare $P\left(x,y\right)$ with the point $\left(-3,-3\right)$ . Substitute $-3$ for $x$ and $-3$ for $y$ in the tangent formula.

  $\begin{array}{c}\tan \theta =\frac{-3}{-3}\\ =1\end{array}$

Use the inverse trigonometric identity to find the angle.

  $\begin{array}{c}\theta ={\tan }^{-1}\left(1\right)\\ \theta =45°\end{array}$

The point lies in the third quadrant. Find the angle in the third quadrant.

  $180°+45°=225°$

Use the conversion factor $\frac{\pi }{180°}$ to convert the angle to radians.

  $225°\times \frac{\pi }{180°}=\frac{5\pi }{4}$

Thus, terminal side of the point $\left(-3,-3\right)$ makes an angle of $225°$ or $\frac{5\pi }{4}$ .