Trigonometry: Exact Values
Master the exact trigonometric values for 30°, 45°, and 60° using special triangles, and explore the unit circle as a foundation for advanced trigonometry.
Why Use Exact Values?
Calculators give decimal approximations (e.g. sin 30° ≈ 0.5000). But in higher mathematics, we often need exact values expressed as fractions or surds. Knowing these allows you to simplify expressions and write proofs without rounding errors.
The Three Special Angles: 30°, 45°, 60°
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | ½ | √3/2 | 1 / √3 |
| 45° | 1 / √2 | 1 / √2 | 1 |
| 60° | √3/2 | ½ | √3 |
Memory tip: for sin, the values follow ½, 1/√2, √3/2 as the angle increases. For cos, they go in reverse order.
Deriving Exact Values from Special Triangles
We derive exact values from two special right-angled triangles. You should be able to draw and use both.
The 45°–45°–90° Triangle
Isosceles right triangle with legs of length 1 and hypotenuse √2
The 30°–60°–90° Triangle
Half an equilateral triangle with hypotenuse 2, short side 1, long side √3
Worked Example: Find sin 60° exactly
Triangle: Use the 30-60-90 triangle: hypotenuse = 2, side opposite 60° = √3
sin 60° = opposite / hypotenuse = √3/2
Introduction to the Unit Circle
The unit circle is a circle with radius 1 centred at the origin. For any angle θ measured from the positive x-axis, the point on the circle has coordinates (cosθ, sinθ).
Worked Example: Evaluate 2 sin 45° cos 45°
Step 1: sin 45° = 1/√2, cos 45° = 1/√2
Step 2: 2 × (1/√2) × (1/√2) = 2 × 1/2 = 1
Worked Example: Simplify sin² 30° + cos² 30°
Step 1: sin 30° = ½, so sin² 30° = ¼
Step 2: cos 30° = √3/2, so cos² 30° = 3/4
Step 3: ¼ + 3/4 = 1
This confirms the Pythagorean identity: sin²θ + cos²θ = 1
Key Vocabulary
Exact Value
A value expressed precisely as a fraction or surd, with no rounding or decimal approximation.
Surd
An irrational number expressed as a root that cannot be simplified to a whole number, e.g. √2 or √3.
Unit Circle
A circle of radius 1 centred at the origin, used to define sine and cosine for all angles.
Pythagorean Identity
The fundamental trig identity: sin²θ + cos²θ = 1 for all angles θ.
Knowledge Check
Test your knowledge of exact trigonometric values.
Question 1
What is the exact value of sin 30°?
Question 2
What is the exact value of tan 45°?
Question 3
What is the exact value of cos 60°?
Question 4
Evaluate exactly: sin 60° × cos 30°
Question 5
On the unit circle, what are the coordinates of the point at angle 30°?
Key Concepts Summary
- ●Exact values avoid rounding errors and are essential in higher mathematics.
- ●45-45-90 triangle gives sin 45° = cos 45° = 1/√2 and tan 45° = 1.
- ●30-60-90 triangle gives sin 30° = cos 60° = ½ and sin 60° = cos 30° = √3/2.
- ●On the unit circle, the point at angle θ has coordinates (cosθ, sinθ).
- ●The Pythagorean identity sin²θ + cos²θ = 1 holds for all angles θ.