Unit Circle Trigonometric Functions

[kyrga]

Homework Statement

I'm trying to do a few problems that ask me to "find the point (x,y) on the unit circle that corresponds to the real number t." Examples of these problems are:
t = pi / 4
t = 7pi / 6
t = 4pi / 3 etc etc

Homework Equations

The Attempt at a Solution

I did a search on this board before posting and found a great cheat sheet, but I was still wondering, is there a formula that allows you to find the point? Or do you just have to memorize/reference a cheat sheet for each angle?

 

[symbolipoint]

Those values for t are very common angles are their corresponding points on the unit circle are well known. Any Trigonometry textbook will clearly show this information. The derivations are taken from simple right triangle Geometry. Note that when the reference angles on the unit circle are multiples of 30 degrees or 45 degrees, finding the x and y values is very simple. You do not need an internet reference site. Mostly pythagorean theorem and Special triangles knowledge is all you need.

 

[HallsofIvy]

Strictly speaking, t here is NOT an angle at all- it measures distance around the unit circle. The unit circle has circumference $2\pi$ so \pi/4 is exactly 1/8 of the entire circle. Since the positive and negative x and y axes divide the circle into 4 equal parts, 1/8 of the circle will be exactly half way between the positive x and y axes and so corresponds to the line y= x. Where does y= x cross the circle x²+ y²= 1?

Another way to get that would be to say that since $\pi/4$ is half of a right angle, if one angle in a right triangle is $\pi/4$ the other angle in a right triangle is also. So the right triangle is an isosceles triangle and the two legs are of equal length. Taking the hypotenuse to be of length 1 and the two legs to be of length x, the Pythagorean theorem, a²+ b²+ c² becomes x²+ x²= 2x²= 1. Solve for x and then remember that the coordinates of the point are (cos(t), sin(t)).

$\pi/3$ is 1/3 of $\pi$ which corresponds to the total 180 degrees in a triangle. A triangle with all angles $\pi/3$ is an equilateral triangle. Take all sides to have length 1. Drop a perpendicular from the vertex of the triangle to the opposite side and you have two right triangles each with hypotenuse of length 1 and one leg of length 1/2. You can use the Pythagorean theorem to find the length of the other leg and then it is easy to find sine and cosine of the angles $\pi/3$ and $\pi/6$ of the right triangle. Again, the point corresponding to t on the unit circle is (cos(t), sin(t)).