Q.45 - 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.

$f_1\left(\theta \right) = {\sin }^2 \theta and f_2\left(\theta \right) = {\cos }^2 \theta$.

Step-by-Step Solution

Answer

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

Part (c) The point of intersection is $\left(\frac{\mathrm{\pi }}{4}, \frac{1}{2}\right) ; \left(-\frac{\mathrm{\pi }}{4}, \frac{1}{2}\right)$.

Part (b)

Part (a) Step 1. Given Information.

The given curve is :

$f_1\left(\theta \right) = {\sin }^2 \theta and f_2\left(\theta \right) = {\cos }^2 \theta$

Part (a) Step 2. Algebraically.

The value of the curve is :

$$\begin{gathered} f_1\left(\theta \right)={\sin }^2\theta , f_2\left(\theta \right) ={\cos }^2\theta \\ {f}_1\left(\theta \right)= f_2\left(\theta \right) \\ {\sin }^2\theta ={\cos }^2\theta \\ \left(\sin \theta + \cos \theta \right)\left(\sin \theta - \cos \theta \right)=1 \\ \tan \theta =-1 , 1 \\ \theta =\pm \frac{\mathrm{\pi }}{4}. \end{gathered}$$

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 curve is :

$$\begin{gathered} f_1\left(\theta \right)={\sin }^2\theta , f_2\left(\theta \right) ={\cos }^2\theta \\ {f}_1\left(\theta \right)= f_2\left(\theta \right) \\ {\sin }^2\theta ={\cos }^2\theta \\ \left(\sin \theta + \cos \theta \right)\left(\sin \theta - \cos \theta \right)=1 \\ \tan \theta =-1 , 1 \\ \theta =\pm \frac{\mathrm{\pi }}{4}. \end{gathered}$$

The point of intersection between the curves are $\left(\frac{\mathrm{\pi }}{4},\frac{1}{2}\right) ; \left(-\frac{\mathrm{\pi }}{4},\frac{1}{2}\right)$.