Please, please, please, go back and review trigonometry! "cos x= 90º" doesn't even make sense: you take the cosine OF an angle, an angle is not the value of a cosine!
If you meant "cos(90º)= 0" that is also wrong. Now, exactly what definition of sine and cosine are you using? One problem with the old "trigonometry" definition of "opposite side over hypotenuse" and "near side over hypotenuse" is that your angles must larger than 0º and less than 90º in order to have a triangle: you can't have an angle in a triangle so that cos(x)= 1 because the near side and hypotenuse cannot be of the same length.
But in higher mathematics, we use sine and cosine as functions for many different applications that have nothing to do with triangles or even angles and we want to be able to define sin(x) and cosine(x) for any number x. A good way to do that is to use the unit circle. Draw a unit circle (center at (0,0), radius 1) on an xy- coordinate system. Starting from the point (1, 0), measure distance t counterclockwise around the circumference of the circle. The coordinates of the end point are, by definition, (cos(t), sin(t)). (If t is negative, measure clockwise.) That is, cos(t) and sin(t) are defined as the x and y coordinates of that point.
Notice that if you draw a line from that end point to the origin and also draw a line from that end point perpendicular to the x-axis, you make a right triangle having hypotenuse of length 1 (because the unit circle has radius 1), near side of length x and opposite side of length y so that cosine and sine of that angle are x/1= x and y/1= y.
But also notice that the "t" in cos(t) and sin(t) here is NOT an angle at all. It is a distance around the circle. If we measure the angle in radians, since radian angle measure is defined as the length of the arc subtending the angle defined by the length, with the unit circle the angle measure in radians is exactly the length of the arc that I am calling "t" here. Degree measure is convenient for working specifically with angles and triangles but for any other application of sine and cosine, we always use "radians".
Of course, since a unit circle with have circumference 2\pi r= 2\pi(1)= 2\pi, we can also see that half a circle, corresponding to 180º, is \pi radians and 1/4 of a circle, corresponding to 90º, is \pi/2 radians.
Now, on that unit circle graph you have drawn, mark the points (1, 0), (0, 1), (-1, 0), (0, -1). That should make it very easy to see what values of t give cos(t) and sin(t) equal to 0, 1, or -1.