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Chapter 9: Problem 20

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=3+\sin \theta \\ r=2 \csc \theta \end{array} $$

Short Answer

Expert verified
The intersection points of the two equations is where `theta = arcsin[(3-sqrt(13))/2]` and `r = 3 + sin(arcsin[(3-sqrt(13))/2])`.

Step by step solution

01

Convert the Equations into Same Trigonometric Function

The second equation can be converted to a sine function equivalent, since csc (theta) = 1 / sin(theta). Therefore, `r = 2 / sin(theta)`.
02

Set up the Equations

Having both equations in terms of sin, set them equal to each other in order to find a common solution. `3 + sin(theta) = 2 / sin(theta)`.
03

Solve the Equation

This equation is a quadratic equation in sin(theta). To make it clear, multiply both sides by sin(theta) to avoid fraction, obtain `sin^2(theta) - 3sin(theta) - 2 = 0`.
04

Solve for Thetas by Applying Quadratic Formula

This is a quadratic equation in sin(theta) and can be solved using quadratic formula sin(theta) = (3卤sqrt(13))/2 . Since \( -1 \leq sin(theta) \leq 1 \), only acceptable answer is sin(theta) = (3-sqrt(13))/2.
05

Calculate Theta Values

The actual thetas can be calculated by taking inverse sine (or arcsin) of the calculated values. The solutions for theta are `theta = arcsin[(3-sqrt(13))/2]`.
06

Solve for the Radius r

Now, plug the values of theta into either of the original formulas to get the r values. For instance, using the first equation, get `r = 3 + sin(arcsin[(3-sqrt(13))/2])`.

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