Question 545782: What is the answer to cot2x-root3=0 ?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  <-->
There are angles that have well known trigonometric functions, and teachers like to give their students tables (or better yet, circular charts) showing those angles and their trigonometric functions. If you do not have one of those, they are easy to build from scratch. They all come from the trigonometric ratios of two types of right triangle: the isosceles right triangle and the 30-60-90 right triangle. I'll tell you more about those below, but the cheat information you need is this:
 , cos(pi/6)=sqrt(3)2}}}
From there you can get all the other trigonometric functions such as:
 and  .
That last one should ring a bell.
30 degrees= is the only angle in the first quadrant with such values for its trigonometric functions.
So  <--> is one solution.
There are infinite other solutions, because cotangent, a periodic function repeats every 180 degrees (or every  ).
So all solutions could be written as
 , with  being any integer.
That gives you
 , with being any integer.
NOW FOR THOSE SPECIAL TRIANGLES
 The 30-60-90 right triangle has angles measuring 30, 60, and 90 degrees, or  ,  , and  in radians.
That triangle is half of an equilateral triangle, with the hypotenuse being one side of the original equilateral triangle, and the short side (the one opposite the 30 degree angle) being half of one side of the original equilateral triangle. Pythagoras theorem tells you that the measure of the third side (the longer leg), in therms of the hypotenuse is:
All 30-60-90 triangles have the same side lengths ratios, which you can use to calculate the exact values of the trigonometry functions for the 30 and 60 degree angles.
The isosceles right triangle is half of a square  and allows you to calculate the exact values for the trigonometric functions for the 45 degree angles ( ).
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