Special & Common Trig Values: Explanation & Overview

Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

For certain angles - called special angles - you can calculate exact trigonometric values. This lesson will discuss these special angles and their common trigonometric values. There will be a quiz at the end of the lesson to solidify your knowledge.

Trigonometric Functions

There are six basic trigonometric functions. They are used mainly to determine either the angles or side lengths of triangles which can be useful in navigation, engineering and physics. Trigonometric functions, especially the sine and cosine functions, are also used to describe things with periodic properties such as light and sound waves.

The six trigonometric functions are:

  • sine (sin)
  • cosine (cos)
  • tangent (tan)
  • cosecant (csc)
  • secant (sec)
  • cotangent (cot)

Values for the trigonometric functions are calculated using the following ratios:

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  • sin = opposite/hypotenuse (opp/hyp)
  • cos = adjacent/hypotenuse (adj/hyp)
  • tan = opposite/adjacent (opp/adj)
  • csc = hypotenuse/opposite (hyp/opp)
  • sec = hypotenuse/adjacent (hyp/adj)
  • cot = adjacent/opposite (adj/opp)

Special Angle Values

There are a few special angles whose trigonometric functions are nice and neat. Because of this, they are the angles most commonly used in calculus problems.

We can find the trigonometric values for these special angles using the above trigonometric ratios.

For example, using the following triangle, we can find the values for 30° and 60°

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  • sin30° = opp/hyp = 1/2
  • cos30° = adj/hyp = √3/2
  • tan30° = opp/adj = 1/√3
  • sec30° = hyp/adj = 2/√3
  • csc30° = hyp/opp = 2/1 = 2
  • cot30° = adj/opp = √3/1 = √3
  • sin60° = opp/hyp = √3/2
  • cos60° = adj/hyp = 1/2
  • tan60° = opp/adj = √3/1
  • sec60° = hyp/adj = 2/1 = 2
  • csc60° = hyp/opp = 2/√3
  • cot60° = adj/opp = 1/√2

And, using this 45-45-90 triangle, we can find the trigonometric functions for a 45° angle.

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  • sin45° = opp/hyp = 1/√2
  • cos45° = adj/hyp = 1/√2
  • tan45° = opp/adj = 1/1 = 1
  • sec45° = hyp/adj = √2/1 = √2
  • csc45° = hyp/opp = √2/1 = √2
  • cot45° = adj/opp = 1/1 = 1

Purpose of Trigonometric Functions for Special Angles

The angles 30°, 45° and 60° are considered to be the most common angles because they are the ones that are seen the most often in real-life situations. For this reason, and because the answers are the 'simplest' of all the trigonometric functions, the trigonometric ratios for these three angles are the most common, and should be memorized.

Memorizing these trigonometric ratios allows you to solve most any trigonometric function with ease. Even without a calculator, the most basic trigonometric problems can be solved.

Examples

1.) Evaluate tan 45°

Answer: For any problem involving a 45°-45°-90° triangle, you shouldn't have to use a table or calculator. You should be able to sketch the triangle and place the ratio numbers.

Since the tangent is the ratio of the opposite side to the adjacent side, you can see that tan 45° = 1/1 = 1

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