polar and rectangular coordinates

Question

The point $(r,\theta)$ in polar coordinates is $(7,5)$ in rectangular coordinates. What is the point $\left( 2r, \theta + \frac{\pi}{2} \right)$ in rectangular coordinates?

Am I supposed to find (7,5) in polar coordinates?
If yes, then how should I solve?

Answer 1

Solving for $r$ and $\theta$ is one way to do it. You can get the answer without actually finding values for $r$ and $\theta$, though.

Consider: replacing $\theta$ with $\theta+\frac{\pi}{2}$ rotates you a quarter turn counter-clockwise. That move sends $(x,y)$ to $(-y,x)$, so we're now at $(-5,7)$. Then, replacing $r$ with $2r$ doubles your distance from the origin, without changing direction. You can do that by doubling both coordinates.

By all means, find the answer by doing trigonometry as well, for practice, but make sure you end up with the same answer that you get this way.

Answer 2

Hint:

Draw a right triangle in Quadrant I with legs of lengths $7$ and $5$. Then, use trigonometry to solve for $r$ and $\theta$. Next, calculate $2r$ and $\theta + \pi/2$. Draw another tight triangle that models these polar coordinates. Now use trigonometry to solve for the lengths of the legs.

This cheat sheet should be of use to you.

Answer 3

We have $$x_1=7=r\cos (\theta)$$ $$y_1=5=r\sin (\theta) $$ and we look for $x_2,y_2$ such that

$$x_2=2r\cos (\theta+\frac \pi 2)$$ $$=-2r\sin (\theta)=-2y_1=-10$$ and $$y_2=2r\sin (\theta+\frac \pi 2)$$ $$=2r\cos (\theta)=14$$