Question

If$f\left(x\right)=\cos \theta =0.3,$findrole="math" localid="1647539382705" $f(\theta +\pi )$

Short answer

The value of$f(\theta +\pi )=-0.3$

Step-by-step solution

Step 1. Given information

$f\left(\theta \right)=\cos \theta =0.3$

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 localid="1647539950698" $x^2+y^2=r^2$.Then

localid="1647539956108" $$\begin{gathered} \sin \theta =\frac{y}{r}\cos \theta =\frac{x}{r}\tan \theta =\frac{y}{x} \\ csc\theta =\frac{r}{y}sec\theta =\frac{r}{x}cot\theta =\frac{x}{y} \end{gathered}$$

Consider $f\left(\theta \right)==\cos \theta =0.3$

That is $$\begin{gathered} \cos \theta =\frac{x}{r} \\ =0.3 \end{gathered}$$

sothat x is positive

therefore,$\theta$is in 1st quadrants

Then $\theta +\pi$ is in the 2nd and 3rd quadrant so that xis negtive

Hence,

localid="1647539987909" $$\begin{gathered} f(\theta +\pi )=\frac{-x}{r} \\ =-\cos \theta \\ =-0.3 \end{gathered}$$