# Finding angle

Find an angle between 0 and 360 degrees for which the ratio of sin to cos is -3. I know this seems to be an easy question, but I am stuck. I appreciate for those helping me.

## Answers and Replies

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Think of the ratio $$\frac{\sin\theta}{\cos\theta}$$. What can that be written as?

I know that can be written as tangent, then?

Consider the quadrants where tan is negative. Then you have to realize you're not going to get a nice angle so you'll have to use the inverse tan function on your calculator. Now I don't know if you have studied the unit circle or rather trigonometry in relation to the coordinate plane, but your calculator will probably spit out an angle between -pi/2 and pi/2, i.e. between -90 degrees and 90 degrees. Using the knowledge of which quadrants tan is negative in, you should be able to figure out the angles between 0 and 360 degrees that correspond to the value your calculator outputs.

well I know that tangent is negative in quadrant 2 and quadrant 4, and using the calculator to find tan-1 (-3.00) I got the result to be -1.249. I know this is not degrees but in radians, am I right? Then from this point, what step do I go next?

Ok good, we're talking in radians. Then the problem is equivalent to finding the angles in the range [0, 2pi] that satisfy tan(x) = -3. Now assuming that you have studied the unit circle, the inverse tan is giving us a negative angle and it is easily seen that $$-\frac{\pi}{2} \leq -1.249 \leq 0$$. This means that we are in the fourth quadrant, and we can imagine a ray starting from the origin that makes an angle of about 1.249 radians with respect to the positive x-axis. Now use symmetry arguments to find out which angles between 0 and 2pi this should correspond to.

use symmetry arguments to find out which angles between 0 and 2pi this should correspond to??

I don't understand that part you said

so first of all I need to convert that radians to degrees right? as I want the final answer in degrees, and how do I do that?

Ok... that's not quite the response I was expecting. Do you understand the kind of analysis used to solve this problem? If your calculator doesn't have a degree mode, you could use the conversion factor $$\frac{\pi}{180\deg} = 1$$.

Ok... that's not quite the response I was expecting. Do you understand the kind of analysis used to solve this problem? If your calculator doesn't have a degree mode, you could use the conversion factor $$\frac{\pi}{180\deg} = 1$$.
OK, I think I figure out the answer it's 288.4 and 108.4, am I right?

Dick