Chapter 6.2, Problem 134AYU

To determine

Explain how to quickly compute the trigonometric functions of $0°,90°,180°,270°$ .

Answer to Problem 134AYU

  $\boxed{\sin 270°=-1,\cos 270°=0,\tan 270°=\boxed{\text{Not defined}},\csc 270°=-1,\sec 270°=\boxed{\text{Not defined}},\cot 270°=0}$

Explanation of Solution

Given information:

Write a brief paragraph that explain how to quickly compute the trigonometric functions of $0°,90°,180°,270°$ .

Calculation:

To find the values of $0°,90°,180°,270°$ , select points $a,b$ , where $a$ is its the distance from $y-axis$ and $b$ is its distance from $x-axis$ . On the terminal side making the angle $0°,90°,180°,270°$ .

Lets choose point $P\left(0,-1\right)$ on the terminal side of $\theta =270°$ and distance,

  $r=1$

Unit from the origin as shown.

  

For example calculate,

  $\begin{array}{l}\sin 270°=\frac{opposite\left(b\right)}{hypotenuse\left(r\right)}\\ =\frac{-1}{1}\\ =-1\end{array}$

  $\begin{array}{l}\cos 270°=\frac{adjacent\left(a\right)}{hypotenuse\left(r\right)}\\ =\frac{0}{1}\\ =0\end{array}$

Similarly calculate,

  $\begin{array}{l}\tan 270°=\frac{\sin 270°}{\cos 270°}\\ =\frac{-1}{0}\\ =not,defined\end{array}$

  $\begin{array}{l}\cot 270°=\frac{\cos 270°}{\sin 270°}\\ =\frac{0}{-1}\\ =0\end{array}$

  $\begin{array}{l}\sec 270°=\frac{1}{\cos 270°}\\ =\frac{1}{0}\\ =not,defined\end{array}$

  $\begin{array}{l}\csc 270°=\frac{1}{\sin 270°}\\ =\frac{1}{-1}\\ =-1\end{array}$

Hence,

  $\boxed{\sin 270°=-1,\cos 270°=0,\tan 270°=\boxed{\text{Not defined}},\csc 270°=-1,\sec 270°=\boxed{\text{Not defined}},\cot 270°=0}$