Chapter 12.3, Problem 50WE

To determine

Give the values of five other trigonometric functions of $\theta$ and find the quadrant.

Answer to Problem 50WE

  $\theta$ is in Quadrant IV.

  $\cos \theta =\frac{\sqrt{21}}{5}$

  $\tan \theta =\frac{-2\sqrt{21}}{21}$

  $\csc \theta =\frac{5}{-2}$

  $\sec \theta =\frac{5\sqrt{21}}{21}$

  $\cot \theta =\frac{\sqrt{21}}{-2}$

Explanation of Solution

Given:

  $\sin \theta =-\frac{2}{5},-90°<\theta <180°$

Formula Used:

  $\sin \theta =\frac{\text{length of side opposite to }\theta }{\text{length of the hypotenuse}}=\frac{y}{r}$

  $\cos \theta =\frac{\text{length of side adjacent to }\theta }{\text{length of the hypotenuse}}=\frac{x}{r}$

  $\tan \theta =\frac{\text{length of side opposite to }\theta }{\text{length of side adjacent to }\theta }=\frac{y}{x}$

  $\csc \theta =\frac{1}{\sin \theta }=\frac{r}{y}$

  $\sec \theta =\frac{1}{\cos \theta }=\frac{r}{x}$

  $\cot \theta =\frac{1}{\tan \theta }=\frac{x}{y}$

Function value	Quadrant of $\theta$ Is I	Quadrant of $\theta$ Is II	Quadrant of $\theta$ Is III	Quadrant of $\theta$ Is IV
$\begin{array}{l}\sin \theta \\ \csc \theta \end{array}$	+	+	-	-
$\begin{array}{l}\cos \theta \\ \sec \theta \end{array}$	+	-	-	+
$\begin{array}{l}\tan \theta \\ \cot \theta \end{array}$	+	-	+	-

Calculation:

  $\sin \theta =-\frac{2}{5},-90°<\theta <180°$

From the table ,

the angle $\theta$ is in Quadrant IV, since sin is negative in it.

  $\sin \theta =\frac{\text{length of side opposite to }\theta }{\text{length of the hypotenuse}}=\frac{y}{r}=\frac{-2}{5}$

So,

x = length of the side adjacent to the angle θ= $\sqrt{r^2-y^2}=\sqrt{25-4}=\sqrt{21}$

y = length of the side opposite to the angle θ= -2

r = length of the hypotenuse = 5

  $\cos \theta =\frac{x}{r}=\frac{\sqrt{21}}{5}$

  $\tan \theta =\frac{y}{x}=\frac{-2}{\sqrt{21}}=\frac{-2\sqrt{21}}{21}$

  $\csc \theta =\frac{r}{y}=\frac{5}{-2}$

  $\sec \theta =\frac{r}{x}=\frac{5}{\sqrt{21}}=\frac{5\sqrt{21}}{21}$

  $\cot \theta =\frac{x}{y}=\frac{\sqrt{21}}{-2}$