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

Before, we can find the sine and cosine, we need to build our 30-60-90 degrees triangle. Start with an equilateral triangle with a side length of 4 like the one you see below.

Equilateral triangle

Then, from one vertex, draw the line that is perpendicular to the side opposite the vertex. Notice that the black line bisect the side. That is why the leg opposite the 30 degrees angle measures 2.

30-60-90 triangle inside an equilateral triangle

All we need to do is to find the length of the leg in black and we will be ready to find sin(30 degrees), sin(60 degrees), cos(30 degrees), and cos(60 degrees).

Call the length of the black line y and use the pythagorean theorem to find y.

42 = 22 + y2

16 = 4 + y2

16 - 4 = 4 - 4 + y2

12 = y2

y = √ (12)

y = √ (4 × 3)

y = √(4) × √(3)

y = 2√3

We pull out the right triangle and we are ready to find the trigonometric values of sine and cosine for this triangle.

30-60-90 degrees triangle
sin(60 degrees) =  
2√3 / 4

sin(60 degrees) =  
√3 / 2

sin(30 degrees) =  
2 / 4

sin(30 degrees) =  
1 / 2

cos(60 degrees) =  
2 / 4

cos(60 degrees) =  
1 / 2

cos(30 degrees) =  
2√3 / 4

cos(30 degrees) =  
√3 / 2

Important observation:

sin(60 degrees) = cos(30 degrees)

cos(60 degrees) = sin(30 degrees)

We will not do this computation, but if we replace 30 degrees with 20 degrees and 60 degrees with 70 degrees, you will find a similar result.

sin(70 degrees) = cos(20 degrees)

cos(70 degrees) = sin(20 degrees)

In general, 

sin(t) = cos(90 degrees - t)

cos(t) = sin(90 degrees - t)

We now have a couple more trigonometric formulas.

Recent Articles

  1. Linear Parent Function

    Nov 20, 17 11:18 AM

    What is a linear parent function? Crystal clear explanation.

    Read More

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

            Follow me on Pinterest

Recent Lessons

  1. Linear Parent Function

    Nov 20, 17 11:18 AM

    What is a linear parent function? Crystal clear explanation.

    Read More

  Our Top Pages

Formula for percentage

Compatible numbers

Basic math test

Basic math formulas

Types of angles

Math problem solver

Algebra word problems

Surface area of a cube

Finding the average 

Scale drawings

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. 

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.


Scientific Notation Quiz

Types of Angles Quiz

Graphing Slope Quiz

Adding and Subtracting Matrices Quiz  

Factoring Trinomials Quiz 

Solving Absolute Value Equations Quiz  

Order of Operations Quiz