Sheet № 164 · Higher only · AQA · Edexcel · OCR
Exact Trigonometric Values –
Exact trigonometric values are a Higher-only GCSE Maths topic that you must learn by heart. AQA, Edexcel, and OCR all expect you to recall the values of sin, cos, and tan for the angles 0°, 30°, 45°, 60°, and 90° without a calculator. This guide lists every value, explains where they come from using special triangles, and gives you worked
§Key definitions
The 45-45-90 triangle:
Take a right-angled isosceles triangle with two short sides of length 1. By Pythagoras, the hypotenuse = sqrt(2). Then sin 45° = 1/sqrt(2) and cos 45° = 1/sqrt(2).
Question:
Write down the exact value of sin 60°.
Answer:
sin 60° = sqrt(3)/2.
Q1 (Higher):
Write down the exact value of tan 60°.
Q2 (Higher):
Find the exact value of 2 sin 30° × cos 30°.
§Formulas to memorise
sin 0° = 0, cos 0° = 1, tan 0° = 0
sin 30° = 1/2, cos 30° = sqrt(3)/2, tan 30° = 1/sqrt(3) = sqrt(3)/3
sin 45° = 1/sqrt(2) = sqrt(2)/2, cos 45° = 1/sqrt(2) = sqrt(2)/2, tan 45° = 1
sin 60° = sqrt(3)/2, cos 60° = 1/2, tan 60° = sqrt(3)
sin 90° = 1, cos 90° = 0, tan 90° = undefined
The 45-45-90 triangle:: Take a right-angled isosceles triangle with two short sides of length 1. By Pythagoras, the hypotenuse = sqrt(2). Then sin 45° = 1/sqrt(2) and cos 45° = 1/sqrt(2).
Identify the angle — in the question — it should be one of 0°, 30°, 45°, 60°, or 90°.
Recall the exact value — from memory or by sketching the appropriate special triangle.
Substitute — into the calculation, keeping values as surds or fractions.
Simplify — the expression, rationalising the denominator if required.
Worked example
Write down the exact value of sin 60°.
This topic is Higher only, but this introductory example uses simple recall.
⚠ Common mistakes
- ✗Confusing sin and cos values for 30° and 60°. sin 30° = 1/2 and cos 30° = sqrt(3)/2, but sin 60° = sqrt(3)/2 and cos 60° = 1/2. The sin and cos values swap between 30° and 60°. A helpful pattern: the larger angle has the larger sin value.
- ✗Forgetting that tan 90° is undefined. The denominator (cos 90° = 0) makes division impossible. Never write tan 90° = infinity — write "undefined".
- ✗Not rationalising the denominator. Some exam boards prefer sqrt(3)/3 over 1/sqrt(3). Practise rationalising: multiply top and bottom by sqrt(3).
✦ Exam tips
- →Create a table of all 15 values (5 angles by 3 functions) and memorise it. Test yourself regularly.
- →Notice the pattern for sin: 0, 1/2, sqrt(2)/2, sqrt(3)/2, 1. This can also be written as sqrt(0)/2, sqrt(1)/2, sqrt(2)/2, sqrt(3)/2, sqrt(4)/2.
- →The cos values are the sin values in reverse order.
- →These values often appear in non-calculator papers alongside surds, Pythagoras, or trigonometry questions. Be ready to substitute exact values instead of reaching for a calculator.
- →Exam questions may ask you to "show that" or "prove" — you must show full working, not just state the answer.