Q. 89 - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(\theta \right)=\sin \theta =0.1$,find $f(\theta +\pi )$

Step-by-Step Solution

Answer

The value of $f(\theta +\pi )=-0.1$

Step 1. Given information

Given function $f\left(\theta \right)=\sin \theta$

Step 2. Finding the value of f ( θ + π )

For an angle $\theta$ is standard position,let $P=(x,y)$ be the point on the terminal side of $\theta$ that is also on the circle.$x^2+y^2=r^2$.Then

$$\begin{gathered} \sin \theta =\frac{y}{r} \cos \theta =\frac{x}{r} \tan \theta =\frac{y}{x} \\ csc\theta =\frac{r}{y} sec\theta =\frac{r}{x} cot\theta =\frac{x}{y} \end{gathered}$$

Consider,

 $$\begin{gathered} f\left(\theta \right)=\sin \theta \\ =0.1 \end{gathered}$$

That is

 $$\begin{gathered} \sin \theta =\frac{y}{r} \\ =0.1 \end{gathered}$$so,that y is positive

Therefore,$\theta$is in 1st or 2nd quadrants.

Then $\theta +\pi$is in 3rd or 4th quadrant so that y is negative

Hence,

$$\begin{gathered} f(\theta +\pi )=-\frac{y}{r} \\ =-\sin \theta \\ =-0.1 \end{gathered}$$

Therefore,the value of $f(\theta +\pi )=-0.1$