Chapter 4, Problem 15RE

Solution

To find: The smallest positive angle measured by the terminal side of the point $\left(6,-12\right)$ in both radians and degrees.

The smallest positive angle made by the terminal side of the point $\left(6,-12\right)$ is $296.57°$ in degrees and $1.65\pi$ in radians.

Given information: The given point is $\left(6,-12\right)$ .

The 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 $\left(6,-12\right)$ . Substitute $6$ for $x$ and $-12$ for $y$ in the tangent formula.

  $\begin{array}{c}\tan \theta =\frac{-12}{6}\\ =-2\end{array}$

Use the inverse trigonometric identity to find the angle.

  $\begin{array}{c}\theta ={\tan }^{-1}\left(-2\right)\\ \theta =-{\tan }^{-1}\left(2\right)\\ \theta =-63.43°\end{array}$

To convert the above angle to a positive value, subtract the angle from $360°$ .

  $360°-63.43°=296.57°$

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

  $296.57°\times \frac{\pi }{180°}=1.65\pi$

Thus, terminal side of the point $\left(6,-12\right)$ makes an angle of $296.57°$ or $1.65\pi$ .