Solving for Sin/Cos Ratio of -3 in [0,360] Degrees

[-EquinoX-]

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.

 

[jeffreydk]

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

 

[-EquinoX-]

I know that can be written as tangent, then?

 

[snipez90]

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.

 

[-EquinoX-]

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?

 

[snipez90]

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.

 

[-EquinoX-]

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

I don't understand that part you said

 

[snipez90]

You need to find the red angle and the green angle. Now you tell me why that will solve the problem.
http://img149.imageshack.us/img149/3258/trig1oo0.th.png

 

[-EquinoX-]

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?

 

[snipez90]

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 $$\frac{\pi}{180\deg} = 1$$.

 

[-EquinoX-]

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]

Yes, those are pretty decent approximations.

 

[HallsofIvy]

You said you got that original -1.249 using a calculator. If that is in radians, then your calculator must be in radian mode. Do you know how to change it to degrees? If your calculator is set to degree mode, then finding arctan(-3) will give you the angle in degrees.