Q. 43. - 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.

$f_1\left(\theta \right)=1+\sin \theta , f_2\left(\theta \right) =1-\cos \theta$.

Step-by-Step Solution

Answer

Part (a)$\theta =\frac{3\mathrm{\pi }}{4}$.

Part (c) $\left(\frac{3\mathrm{\pi }}{4},1.707\right).$

Part (b) 

Part (a) Step 1. Given Information.

The given curves are :

$f_1\left(\theta \right)=1+\sin \theta , f_2\left(\theta \right) =1-\cos \theta$.

Part (a) Step 2. Algebraically.

The solution is :

$$\begin{gathered} f_1\left(\theta \right)=1+\sin \theta , f_2\left(\theta \right) =1-\cos \theta \\ {f}_1\left(\theta \right)= f_2\left(\theta \right) \\ 1+\sin \theta =1-\cos \theta \\ \sin \theta =-\cos \theta \\ -\tan \theta =1 \\ \tan \theta =-1 \\ \theta =\frac{3\mathrm{\pi }}{4}. \end{gathered}$$

Part (b) Step 2. Graphing calculator.

Consider the given information in part (a).

The graph is :

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

Consider the given information from part (a).

The point of intersection of the curves are :

$$\begin{gathered} f_1\left(\theta \right)=1+\sin \theta , f_2\left(\theta \right) =1-\cos \theta \\ {f}_1\left(\theta \right)= f_2\left(\theta \right) \\ 1+\sin \theta =1-\cos \theta \\ \sin \theta =-\cos \theta \\ -\tan \theta =1 \\ \tan \theta =-1 \\ \theta =\frac{3\mathrm{\pi }}{4}. \\ \theta =\frac{3\mathrm{\pi }}{4}+n\mathrm{\pi } \\ \mathrm{points} \mathrm{are} \left(\frac{3\mathrm{\pi }}{4}, 1.707\right) \mathrm{and} \mathrm{so} \mathrm{on} \mathrm{by} \mathrm{substituting} \mathrm{the} \mathrm{values}. \end{gathered}$$