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 examples and practice questions to cement the knowledge.

What Are Exact Trigonometric Values?

Instead of using a calculator, you must know the exact (surd or fraction) values of sin, cos, and tan for five key angles. These values come from two special triangles: the 45-45-90 isosceles right-angled triangle and the 30-60-90 triangle (half of an equilateral triangle).

Key Formulas

Where Do These Values Come From?

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).

The 30-60-90 triangle: Take an equilateral triangle with side length 2 and cut it in half vertically. The resulting right-angled triangle has a hypotenuse of 2, a short side of 1, and a remaining side of sqrt(3) (by Pythagoras). This gives sin 30° = 1/2, cos 30° = sqrt(3)/2, sin 60° = sqrt(3)/2, and cos 60° = 1/2.

Step-by-Step Method

1. Identify the angle in the question — it should be one of 0°, 30°, 45°, 60°, or 90°.
2. Recall the exact value from memory or by sketching the appropriate special triangle.
3. Substitute into the calculation, keeping values as surds or fractions.
4. Simplify the expression, rationalising the denominator if required.

Worked Example 1 — Foundation Level

This topic is Higher only, but this introductory example uses simple recall.

Question: Write down the exact value of sin 60°.

Working:

From the 30-60-90 triangle, sin 60° = sqrt(3)/2.

Answer: sin 60° = sqrt(3)/2.

Worked Example 2 — Higher Level

Question: Find the exact value of sin 30° + cos 60°.

Working:

sin 30° = 1/2

cos 60° = 1/2

sin 30° + cos 60° = 1/2 + 1/2 = 1

Answer: sin 30° + cos 60° = 1.

Worked Example 3 — Exam Style

Question: Show that sin²45° + cos²45° = 1.

Working:

sin 45° = sqrt(2)/2, so sin²45° = (sqrt(2)/2)² = 2/4 = 1/2

cos 45° = sqrt(2)/2, so cos²45° = (sqrt(2)/2)² = 2/4 = 1/2

sin²45° + cos²45° = 1/2 + 1/2 = 1

Answer: Verified. sin²45° + cos²45° = 1.

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.

Practice Questions

Q1 (Higher): Write down the exact value of tan 60°.

Q2 (Higher): Find the exact value of 2 sin 30° × cos 30°.

Q3 (Higher): A right-angled triangle has a hypotenuse of 10 cm and one angle of 30°. Find the exact length of the side opposite the 30° angle.

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Related Topics

• Trigonometry — SOHCAHTOA
• Surds
• Rationalising the Denominator

Summary

• You must memorise the exact values of sin, cos, and tan for 0°, 30°, 45°, 60°, and 90°. These values come from two special triangles: the 45-45-90 isosceles triangle and the 30-60-90 half-equilateral triangle. The sin values follow the pattern sqrt(0)/2 through sqrt(4)/2, and the cos values are the same sequence reversed. Tan 90° is undefined. These values appear frequently on non-calculator Higher papers, often combined with surds or proof-style questions.