Trigonometric Ratios for Special Angles
Simple Explanation
The angles 0°, 30°, 45°, 60°, and 90° have exact, memorizable trigonometric ratios, derived directly from the 45-45-90 and 30-60-90 special right triangles (Chapter 8) — no calculator or decimal approximation needed.
Why Do We Need It?
These exact values appear constantly throughout trigonometry and calculus — memorizing them (or knowing how to quickly re-derive them from the special triangles) saves significant time.
See It
A circle with three points marked at 30, 45, and 60 degrees, each connected to the centre by a radius
Formula
Trigonometric Ratios for Special Angles
sin: 0, 1/2, √2/2, √3/2, 1 | cos: 1, √3/2, √2/2, 1/2, 0 (for θ = 0°, 30°, 45°, 60°, 90°)
The sine and cosine of the five most common "special" angles have exact, memorizable values, derived directly from the 45-45-90 and 30-60-90 triangles.
- θ
- — one of the five special angles: 0°, 30°, 45°, 60°, or 90°
When to use it: Whenever a problem involves one of these five common angles and you need an exact value rather than a decimal approximation.
Worked Example
Derive sin(30°) and cos(30°) from a 30-60-90 triangle
Using a 30-60-90 triangle with sides x, x√3, 2x, find sin(30°) and cos(30°).
Why Does This Work?
Since the special right triangles have fixed, known side ratios (independent of size), plugging those exact ratios into the definitions of sine and cosine (opp/hyp, adj/hyp) always produces the same exact values — no approximation is ever needed.
Real-Life Example
Exact calculations in engineering design
An engineer designing a component with a 60° angle wants an exact, not approximate, value for a related trigonometric ratio, to avoid compounding rounding errors.
Using the special angle exact values (like cos(60°) = 1/2) keeps subsequent calculations perfectly precise, rather than accumulating decimal rounding errors.
Practice
What is tan(45°)?
MediumCommon mistake
Mixing up sin and cos values for 30° and 60° — they are "swapped" (sin30°=cos60°=1/2, cos30°=sin60°=√3/2) since 30° and 60° are complementary angles.
Quick Review
- sin: 0, 1/2, √2/2, √3/2, 1 for θ = 0°, 30°, 45°, 60°, 90° (cos is the same list reversed).
- Derive these from the 45-45-90 and 30-60-90 special right triangles.
- 30° and 60° are complementary, so their sin and cos values swap.