# Thread: getting wrong theta from tan(theta)=-4/3

1. ## getting wrong theta from tan(theta)=-4/3

The exercise asks, convert (-3,4) to polar coordinates.
I get (5, -4.140), but it's wrong. The solution is supposed to be (5, 2.21) or (-5, 5.36).

Why do I keep getting the wrong answer?

Thanks,
Ivy

2. (-5, 5.36) could not possibly be right, a radius length can only ever be positive...

Anyway, $\displaystyle r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

Now, notice that (-3, 4) is in the second quadrant, so $\displaystyle \theta = \pi - \arctan{\left(\frac{4}{3}\right)}$.

3. Hello, Ivy!

Sorry, Prove it . . . $r$ can be negative.

Convert (-3,4) to polar coordinates.

I get (5, -4.140), but it's wrong.
The solution is supposed to be (5, 2.21) or (-5, 5.36).

Why do I keep getting the wrong answer?
. . How did you get that angle?
Code:
       (-3,4) |
*   |
:\  |
4: \5|
:  \|
- - + - + - - - -
3 |

We see that: . $r \,=\,5.$

The angle is: . $\theta \:=\:\tan^{-1}\left(\text{-}\tfrac{4}{3}\right) \:\approx\:\text{-}0.927\text{ radians}$

This translates to positive angles of: . $2.21\text{ and }5.36\text{ radians.}$

You should be aware that there are an infinite number of ways
. . to designate a point in polar coordinates.

Two of the ways are: . $(2.21,\,5)$ and $(-5,\,5.36)$

Recall how polar coordinates are plotted.

The first $(5,\,2,21)$ tells us:
. . stand at the pole (origin), face East, turn 2.21 radians CCW
. . and walk forward 5 units.
This places us at the point in Quadrant 2.

The second $(-5,\,5.36)$ tells us:
. . stand at the pole, face East, turn 5.36 radians CCW
. . and walk backward 5 units.
This places us at the same point.

4. Thank you so much. I get confused about when to use my calculator in radians or degrees mode. Do polar coordinates require that theta always be in rads instead of degrees? When I take the trig. ratio of two sides, will the answer always be in rads?