Q. 41 - 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. $f_1\left(\theta \right) = \sin \theta and f_2\left(\theta \right) = \cos \theta$.

Step-by-Step Solution

Answer

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

Part (c) $\left(\frac{\pi }{4},0.707\right) ; \left(\frac{5\mathrm{\pi }}{4},-0.707\right),....$

Part (b) 

Part(a) Step 1. Given Information.

The given curves are :

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

Part (a) Step 2. Algebraically calculation.

Consider the given information from part (a).

When

$$\begin{gathered} f_1\left(\theta \right)=f_2\left(\theta \right) \\ \sin \left(\theta \right)=\cos \left(\theta \right) \\ \sin \left(\theta \right) - \cos \left(\theta \right) =0 \\ \frac{\sin \theta }{\cos \theta } =1 \\ \tan \theta =1 \\ \tan \theta =\tan \left(\frac{\mathrm{\pi }}{4}\right) \\ \theta =\frac{\pi }{4}. \end{gathered}$$

Part (b) Step 3. Sketching of the two curves.

Consider the given information from part (a).

The curves are :

Part (c) Step 4. Point of Intersection.

Consider the given information from part (a).

The point of intersection of two curves are :

$$\begin{gathered} f_1\left(\theta \right)=f_2\left(\theta \right) \\ \sin \left(\theta \right)=\cos \left(\theta \right) \\ \sin \left(\theta \right) - \cos \left(\theta \right) =0 \\ \frac{\sin \theta }{\cos \theta } =1 \\ \tan \theta =1 \\ \tan \theta =\tan \left(\frac{\mathrm{\pi }}{4}\right) \\ \theta =\frac{\pi }{4}. \end{gathered}$$

The point of intersection of the two curves are $\left(\frac{\pi }{4},0.707\right) ; \left(\frac{5\mathrm{\pi }}{4},-0.707\right),....$