Why do some trigonometric functions have negative values?

Question

Hello,

I am struggling with my trigonometry homework and can't find anything particularly helpful on the internet. For instance, we were asked to find the value of x on the unit circle for the following equation:

sin^2(x)+5cos^2(x) = 3

I managed to do the usual algebraic manipulation to arrive at sin(x) = 1/square root of 2 and then used arcsin to arrive at 45 degrees. From there I used the 1st and 2nd quadrant, as the answer is positive, to arrive at 45 degrees and 135 degrees for x. My book, however, also gives -45 and -135 as answers and I can't understand why.

I am not sure if it is related but we were also asked to find, as above, the value of x between 0 and 360 degrees for cot(2x) = -2/5. I can get 149.9 and 325.9 degrees but the teacher also gets 55.9 and 235.9 degrees and again, I can not see why. In this case the latter two figures seem to be measured from the y coordinate. There is clearly something I am missing out and no where on the internet can I find what it is.

Any help would be greatly appreciated.

Thanks in advance,

Christian B.

Answer 1

sin² x + 5cos² x = 3

sin² x + 5(1 - sin² x) = 3

sin² x + 5 - 5 sin² x = 3

-4 sin² x = -2

sin² x = 1 / 2

sin x = ±√2 / 2 (See the ±?)

For the positive value, 45° and 135°.

For the negative value, -45° (or 315°) and -135° (or 225°)