Q. 44. - 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.

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

Step-by-Step Solution

Answer

Part (a)

$\theta =\frac{\mathrm{\pi }}{3}$

Part (c) Point of intersection is $\left(\frac{\mathrm{\pi }}{3},0.866\right).$

Part (b) 

Part (a) Step 1. Given Information.

The given curves are 

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

Part (a) Step 2. Algebraically.

The solution is :

$$\begin{gathered} f_1\left(\theta \right)=\sin \theta , f_2\left(\theta \right) =\sin 2\theta \\ {f}_1\left(\theta \right)= f_2\left(\theta \right) \\ \sin \theta =\sin 2\theta \\ \sin \theta =2 \sin \theta \cos \theta \\ \cos \theta =\frac{1}{2} \\ \theta =\frac{\mathrm{\pi }}{3}. \end{gathered}$$

Part (b) Step 2. Graphically.

Consider the given information from part (a).

The graph of the given curve is :

Part (c) Step 2. Point of intersection.

Consider the given information from part (a).

The point of intersection of the two curves are :

$$\begin{gathered} f_1\left(\theta \right)=\sin \theta , f_2\left(\theta \right) =\sin 2\theta \\ {f}_1\left(\theta \right)= f_2\left(\theta \right) \\ \sin \theta =\sin 2\theta \\ \sin \theta =2 \sin \theta \cos \theta \\ \cos \theta =\frac{1}{2} \\ \theta =\frac{\mathrm{\pi }}{3}. \end{gathered}$$

The point of intersection is $\left(\frac{\mathrm{\pi }}{3},0.866\right)$.