Question

Please help with geometry

Answer 1

The way to do this is to set up a 30 degree angle in a coordinate plane in the first quadrant.  I say the first quadrant since the first quadrant goes from 0 to 90 degrees and 30 falls right in that interval.  Using the positive x-axis as the initial ray of the 30 degree angle and the terminal ray of the angle as the hypotenuse of a right triangle, if we drop a height from the end of the terminal ray to the x-axis we have formed said right triangle.  The angle at the origin is the 30 degree angle.  According to the Pythagorean triple for a 30-60-90 triangle, the side across from the 30 degree angle measures 1, which is the height of our triangle.  The side across from the 60 degree angle is square root of 3, which is the base of our triangle, and the hypotenuse is 2.  The cos identity is the ratio that utilizes the side adjacent to the reference angle over the hypotenuse, which for us is $\frac{ \sqrt{3} }{2}$.    That's the third choice down.  Finding an "exact" value means that they want you to NOT express your answer in decimal form.