Trig Ratios for multiples of 30, 45 and 60 degrees

Examples, solutions, and videos to help GCSE Maths students learn how to find the Trig Ratios for multiples of 30, 45 and 60 degrees.

The following diagram shows the trig ratios of special angles: 0°, 30°, 45°, 60°, and 90°. Scroll down the page for more examples and solutions on the trigonometric ratios.

Trigonometry Worksheets
Practice your trigonometry skills with the following worksheets.

• Trigonometry Worksheets

  Printable
  Trig Ratios: Sin, Cos, Tan
  Sin & Cos of Complementary Angles
  Find Missing Sides
  Find Missing Angles
  Area of Triangle using Sine
  Law of Sines and Cosines
  

  Online
  Trigonometry (sine, cosine, tangent)
  Trigonometry (sine, cosine, tangent)
  Trigonometry (using a calculator)
  Inverse Trigonometry (using a calculator)
  Trigonometry (find an unknown side)
  Trigonometry (find an unknown angle)
  
  Using Sine
  Using Cosine
  Using Tangent
  Using Sine, Cosine or Tangent
  
  Trigonometry Applications Problems
  

How to remember the Trig Ratios?
There are three main methods to help you memorize the special trig ratios:

1. Patterns in the table
   Notice the following patterns in the above table.
   Sine:
   Think of the numerator as 0, √1, √2, √3
   The denominator is always 2.
   Simplified: 0, 1/2, √2/2, √3/2
   Cosine:
   Think of the numerator as √3, √2, √1, 0
   The denominator is always 2.
   Simplified: √3/2, √2/2, 1/2, 0
   Tangent:
   Use the relationship
   tan(θ)=sin(θ)/cos(θ)
   Simplified: 0, 1/√3, 1, √3, Undefined
   

Have a look at the following video.

• Patterns in the Trig Ratio Table

2. Use special triangles
   Use the 45°-45°-90° Triangle and the 30°-60°-90° Triangle

Have a look at the following lesson for explanations on the special triangles.
Trig Special Triangles

The following video will show you how to calculate the exact values of sin, cos and tan of 45 degrees using the 45°-45°-90° Triangle.

• Show video

The following video will show you how to calculate the exact values of sin, cos and tan of 30 and 60 degrees using the 30°-60°-90° Triangle.

• Show video

3. Use “Hand Trick”
   The following video will show you how to find the trig ratios using the hand trick.
   
   • Show video

Trig. Ratios for positive multiples 30, 45 and 60 degrees
In this tutorial I show you how to calculate the exact values of sin, cos and tan for positive multiples of 30 and 60 degrees.

• Show Step-by-step Solutions

Trig. Ratios for negative multiples 30, 45 and 60 degrees
In this tutorial I show you how to calculate the exact values of sin, cos and tan for negative multiples of 30 and 60 degrees.

• Show Step-by-step Solutions

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