For students and parents · O-Level A-Math
Special Angles, Exact Values Without a Calculator
Exact trig values for 30, 45 and 60 degrees, without a calculator.
Mrs Eileen Toh
Founder & Curriculum Architect · ex-MOE · 3 min read · Updated 29 Jun 2026
A-Math will ask for the exact value of a trig ratio, and a decimal off the calculator will not earn the mark. The special angles, 30, 45 and 60 degrees, have exact values in surd form, and the syllabus expects you to know them and to keep your working in that form to the end.
Students who memorise a table of nine numbers tend to forget one under pressure and have nothing to fall back on. GPA teaches the values from two triangles instead, because a triangle you can redraw in ten seconds is a value you can never truly lose.
The two triangles
The 45-45-90 triangle. Take a right-angled triangle with both short sides equal to 1. By Pythagoras the hypotenuse is √(2). Reading off the 45 degree angle:
sin 45 = 1/√(2), cos 45 = 1/√(2), tan 45 = 1.
The 30-60-90 triangle. Take an equilateral triangle of side 2 and cut it in half. You get a right-angled triangle with hypotenuse 2, short side 1 (half the base) and longer side √(3) by Pythagoras. Reading off both angles gives every remaining value.
The values, from the triangles
| Angle | sin | cos | tan |
|---|---|---|---|
| 30 | 1/2 | sqrt(3)/2 | 1/sqrt(3) |
| 45 | 1/sqrt(2) | 1/sqrt(2) | 1 |
| 60 | sqrt(3)/2 | 1/2 | sqrt(3) |
Each one checks against its decimal: cos 30 = 0.8660..., which is √(3)/2; tan 60 = 1.7320..., which is √(3); sin 30 is exactly 0.5. If a value ever slips in the exam, redraw the matching triangle and read it back off. You do not need the table; you need the two triangles.
Why the exact form matters to the mark
Two habits cost students marks here, and both are avoidable.
The first is decimalising too early. If a later step squares or adds these values, a rounded decimal drifts and the final answer lands outside the accepted range. Carrying √(3)/2 to the end keeps the answer exact and safe.
The second is leaving a surd in the denominator when the form expects it rationalised. 1/√(2) is often written √(2)/2, and 1/√(3) as √(3)/3. Knowing the question's expected form is part of knowing the value.
The teaching point
The special angles are not a memory test; they are a two-triangle test. A student who owns the 45-45-90 and 30-60-90 triangles can rebuild any of the nine values on demand and will keep working in exact form because the triangle hands it to them in exact form. That is the small, reliable foundation under a large amount of trigonometry, from the R-formula to solving triangles.
If your child reaches for the calculator on a question that asked for an exact value, that is a habit to retrain, not a sign of weakness. GPA's Secondary Math programme builds these foundations from the structure up, on real A-Math papers. A short diagnostic consult will show you which habits are leaking marks.
Every value on this page was checked against its decimal before publishing.