Q. 42 - Calculus

Question

For each pair of functions in Exercises 40–45, (a) Algebraically find all values of $\theta$ where $f_1\left(\theta \right)=f_2\left(\theta \right)$. (b) Sketch the two curves in the same polar coordinate system. (c) Find all points of intersection between the two curves.

$f_1\left(\theta \right)=1 , f_2\left(\theta \right)=2 \sin 2\theta$.

Step-by-Step Solution

Answer

Part (a) $\theta =\frac{\mathrm{\pi }}{12}$.

Part (c) $\theta =\frac{\mathrm{\pi }}{12}+n\mathrm{\pi }, \mathrm{\theta }=\frac{5\mathrm{\pi }}{12}+\mathrm{n\pi }.$

Part (b) 

Part(a) Step 1. Given Information.

The given curves are :

$f_1\left(\theta \right)=1 , f_2\left(\theta \right)=2 \sin 2\theta$.

Part (a) Step 2. Algebraically.

We can calculate this by,

$$\begin{gathered} f_1\left(\theta \right)=1 , f_2\left(\theta \right)=2 \sin 2\theta \\ {f}_1\left(\theta \right)= f_2\left(\theta \right) \\ 1=2 \sin 2\theta \\ \frac{1}{2}= \sin 2\theta \\ \frac{\mathrm{\pi }}{6}=2\theta \\ \theta =\frac{\mathrm{\pi }}{12} \end{gathered}$$

Part (b) Step 2. Graphing calculator.

Consider the given information from part (a).

The graph of the curves are :

Part (c) Step 2. Point of intersection of the curves.

Consider the given information from part (a).

The point of intersection is 

$$\begin{gathered} f_1\left(\theta \right)=1 , f_2\left(\theta \right)=2 \sin 2\theta \\ {f}_1\left(\theta \right)= f_2\left(\theta \right) \\ 1=2 \sin 2\theta \\ \frac{1}{2}= \sin 2\theta \\ \frac{\mathrm{\pi }}{6}=2\theta \\ \theta =\frac{\mathrm{\pi }}{12} \\ \theta =\frac{\mathrm{\pi }}{12}+n\pi , \frac{5\mathrm{\pi }}{12}+n\pi \end{gathered}$$.