Trigonometric Ratios of Special Angles: 0°, 30°, 45°, 60°, 90°

How To Derive And Memorize The Trigonometric Ratios Of The Special Angles: 30°, 45° And 60°?

The following special angles chart show how to derive the trig ratios of 30°, 45° and 60° from the 30-60-90 and 45-45-90 special triangles. Scroll down the page if you need more examples and explanations on how to derive and use the trig ratios of special angles.

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1. Using 45°-45°-90° Special Right Triangle:
   Consider a right-angled isosceles triangle with two equal sides of length 1. By the Pythagorean theorem, the hypotenuse has a length of \(\sqrt{1^2 + 1^2} = \sqrt{2}\).
   The angles are 45°, 45°, and 90°.
   Using the definitions of trigonometric ratios (SOH CAH TOA):
   sin(45°) = \(\frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
   cos(45°) = \(\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
   tan(45°) = \(\frac{\text{opposite}}{\text{adjacent}} = \frac{1}{1} = 1\)
   

2. Using 30°-60°-90° Special Right Triangle:
   The sides of this triangle will be hypotenuse = 2, side opposite 30° = 1, and side opposite 60° = \(\sqrt{2^2 - 1^2} = \sqrt{3}\)
   Using the definitions of trigonometric ratios (SOH CAH TOA):
   sin(30°) = \(\frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}\)
   cos(30°) = \(\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}\)
   tan(30°) = \(\frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)
   sin(60°) = \(\frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}\)
   cos(60°) = \(\frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{2}\)
   tan(60°) = \(\frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3}\)
   

Trigonometric Function Values Of Special Angles

How to derive the trigonometric function values of 30, 45 and 60 degrees and their corresponding radian measure. Cofunction identities are also discussed:
sin θ = cos(90° - θ)
cos θ = sin(90° - θ)

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How To Find Trig Ratios Of Special Angles?

This video shows how to find the trig ratios of the special angles: sine, cosine and tangent values of 0, 30, 45, 60 and 90 degrees. Scroll down the page for part 2.

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How To Use The Trig Ratios Of Special Angles To Find Exact Values Of Expressions?

How to use the trig ratios of special angles to find exact values of expressions involving sine, cosine and tangent values of 0, 30, 45, 60 and 90 degrees?

Example:
Determine the exact values of each of the following:
a) sin30°tan45° + tan30°sin60°
b) cos30°sin45° + sin30°tan30°

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How To Remember The Trig Ratios For Special Angles?

A Finger Trick for Trigonometry
If we insist that students memorize the values of sine and cosine for the basic angles 0, 30, 45, 60 and 90 degrees, then here’s a cute little trick for doing so using the fingers on your hand.

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Cool Pattern For Trig Special Angles

A pattern to help you remember the Sine and Cosines of Special Angles in the first quadrant. 0, 30, 45, 60, 90 degrees.

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How To Evaluate Trig Functions Of Special Angles?

Easy way to use right triangle and label sides to find sin, cos, tan, cot, csc, and sec of the special angles, and of angles at multiples of 90°. This is Part 1. Scroll down the page for part 2.

Example:
Find cos 90, tan 90, sin 630, sin 135, tan (-405), sin 210, tan (-30).

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Trigonometric Functions Of Special Angles, Part 2

Example:
Find cos 300, cot 180, sin 1305, sec(-210), csc 750, cos 270, sin(-420).

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