What you'll learn
This topic covers the exact trigonometric values for 0°, 30°, 45°, 60° and 90° — values you are expected to know without a calculator. In this guide you will learn the exact values of sin, cos and tan at these angles, how to remember them, where they come from, and how to use them in non-calculator questions. These are essential higher-tier facts.
Key terms and definitions
Exact value — a value given precisely, often as a surd or fraction, not a decimal.
Surd — a root left in exact form (e.g. √3).
Special angles — 0°, 30°, 45°, 60° and 90°.
Unit fraction surd — values like √3/2 or 1/√2.
Rationalising — removing a surd from a denominator.
Core concepts
The values to know
You must know these exact values:
- sin: 0° = 0, 30° = ½, 45° = √2/2, 60° = √3/2, 90° = 1.
- cos: 0° = 1, 30° = √3/2, 45° = √2/2, 60° = ½, 90° = 0.
- tan: 0° = 0, 30° = 1/√3, 45° = 1, 60° = √3, 90° = undefined.
Spotting the patterns
For sin, the values 0°→90° follow √0/2, √1/2, √2/2, √3/2, √4/2 — a neat pattern. Cos is sin reversed (cos 30° = sin 60°). tan = sin ÷ cos, which is why tan 45° = 1 and tan 90° is undefined (division by zero).
Where they come from
The 45° values come from a right-angled isosceles triangle (sides 1, 1, √2). The 30° and 60° values come from an equilateral triangle cut in half (sides 1, 2, √3). Drawing these triangles helps you recover the values if you forget.
Using them without a calculator
In non-calculator questions, substitute the exact value and simplify, often leaving a surd in the answer. For example, in a right-angled triangle with a 30° angle and hypotenuse 8, the opposite side is 8 × sin 30° = 8 × ½ = 4.
Rationalising surd answers
Answers like 1/√3 are usually rationalised to √3/3 by multiplying top and bottom by √3. Present exact answers tidily, with no surd in the denominator.
Worked examples
Example 1: Using sin 30°
Find the opposite side when the hypotenuse is 10 and the angle is 30°.
10 × sin 30° = 10 × ½ = 5.
Example 2: Using tan 60°
Find the opposite side when the adjacent is 4 and the angle is 60°.
4 × tan 60° = 4√3.
Example 3: cos 45°
State the exact value of cos 45°.
√2/2 (equivalently 1/√2).
Common mistakes and how to avoid them
Mixing up sin and cos. Remember cos is sin reversed across the angles.
Giving decimals. Leave answers exact, as surds or fractions.
Forgetting tan 90° is undefined. Division by cos 90° = 0.
Leaving a surd in the denominator. Rationalise to tidy the answer.
Guessing the values. Reconstruct them from the two special triangles.
Exam technique for Exact Trig Values
Learn the table of sin, cos and tan for the special angles.
Use the √n/2 pattern for sin (and reverse for cos).
Recall the two triangles if you forget a value.
Substitute exact values and keep surds in the answer.
Rationalise any surd denominators.
Quick revision summary
You must know the exact trig values for the special angles. For sin: 0, ½, √2/2, √3/2, 1 (at 0°, 30°, 45°, 60°, 90°), following the pattern √0/2 … √4/2. Cos is sin reversed (cos 30° = √3/2 = sin 60°), and tan = sin ÷ cos, giving tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, with tan 90° undefined. These come from two triangles: the right-angled isosceles (1, 1, √2) gives the 45° values, and the half-equilateral (1, 2, √3) gives the 30° and 60° values. In non-calculator work, substitute the exact value, keep surds in the answer, and rationalise any surd denominators (1/√3 → √3/3). The common errors are mixing sin and cos, giving decimals, forgetting tan 90° is undefined, and leaving surds in denominators. Learn the table, use the √n/2 pattern, reconstruct from the triangles, and present tidy exact answers.