Using the unit circle to find cosine and sine of 30 degrees and 60 degrees

Recommended Lessons

Not sure where to start? These are the lessons and games we recommend most to students, parents and teachers on Basic-mathematics.com. Each lesson explains the topic in plain language, walks you through worked examples step by step, and ends with practice so you can check your understanding. Pick one below, or use the menu to find lessons on fractions, algebra, geometry, statistics and more.

Since we are using the unit circle, we need to put the 30-60-90 triangle inside the unit circle.

Unit circle and the 30-60-90 degrees triangle


The radius of the circle is also the hypotenuse of the right triangle and it is equal to 1.

We have already seen in the previous lesson that the leg opposite the 30 degrees angle is half the hypotenuse. 

Again, we will find the length of vertical black line of the triangle using the pythagorean theorem.

Since 1/2 = 0.5, we will replace 1/2 with 0.5 in the formula to simplify the computation.

12 = 0.52 + y2

1 = 0.25 + y2

1 - 0.25 = 0.25 - 0.25 + y2

0.75 = y2

Since 0.75 = 3/4,   y = √ (3/4)

y = √(3) / √(4)

y = √(3) / 2

x = 1/2, y = √3 / 2 and t could be 30 or 60 degrees.

cos(30 degrees) = y / 1 = y = √3 / 2

cos(60 degrees) = x / 1 = x = 1 / 2

sin(60 degrees) = y / 1 = y = √3 / 2

sin(30 degrees) = x / 1 = x = 1 / 2

Using the 30-60-90 degrees triangle to find sine and cosine