Exact Trigonometric Values: GCSE Maths Guide (30°, 45°, 60°)

Exact trigonometric values are the precise surd or fraction forms of sin, cos, and tan at five special angles — 0°, 30°, 45°, 60°, and 90° — that every GCSE student must know. Whenever an exam says "leave your answer in surd form" or "without a calculator", these exact values are the key.

What are exact trigonometric values and why do they matter?

When you press sin 30° on a calculator, you see 0.5 — a clean, exact result. But sin 45° returns something like 0.7071067…, a non-terminating decimal. Using that rounded figure in further calculations introduces errors. The exact value, √2/2, introduces no such error.

Exact trigonometric values are precise expressions — fractions or surds — for sin, cos, and tan at five special angles: 0°, 30°, 45°, 60°, and 90°. When a GCSE question says "without a calculator" or "give your answer in surd form", that is a direct instruction to use these values. Writing a decimal approximation in those questions will cost marks even if the rest of your working is perfect.

These values also appear in proof questions (such as verifying that sin²θ + cos²θ = 1 for a specific angle), in finding exact side lengths of triangles, and throughout A-level mathematics — so mastering them now pays dividends for years.

How do I derive the exact values for 30° and 60°?

Start with an equilateral triangle with all sides of length 2. All three interior angles are 60°.

Draw the perpendicular from one vertex to the midpoint of the opposite side. This creates two congruent right-angled triangles. In each right-angled triangle:

• Hypotenuse = 2 (one full side of the equilateral triangle)
• Base = 1 (half the bottom side)
• Height = √(2² − 1²) = √(4 − 1) = √3 (by Pythagoras' theorem)

Now read off the trigonometric ratios using SOHCAHTOA.

For the 60° angle (at the top):

• sin 60° = opposite ÷ hypotenuse = √3 ÷ 2 = √3/2
• cos 60° = adjacent ÷ hypotenuse = 1 ÷ 2 = 1/2
• tan 60° = opposite ÷ adjacent = √3 ÷ 1 = √3

For the 30° angle (at the bottom), opposite and adjacent swap:

• sin 30° = 1/2
• cos 30° = √3/2
• tan 30° = 1/√3 = √3/3 (rationalised form)

Notice: sin 30° = cos 60° = 1/2, and sin 60° = cos 30° = √3/2. The sine and cosine of complementary angles always swap — this is a useful memory check.

How do I derive the exact values for 45°?

Start with an isosceles right-angled triangle with both shorter sides (the legs) of length 1.

By Pythagoras' theorem: hypotenuse = √(1² + 1²) = √2

Both acute angles are 45°. Reading off SOHCAHTOA for either 45° angle:

• sin 45° = opposite ÷ hypotenuse = 1 ÷ √2 = √2/2 (after rationalising)
• cos 45° = adjacent ÷ hypotenuse = 1 ÷ √2 = √2/2
• tan 45° = opposite ÷ adjacent = 1 ÷ 1 = 1

Because both legs are equal, sin 45° and cos 45° are identical — both equal √2/2 — and tan 45° = 1 exactly. These are the easiest values to remember.

What is the complete table of exact trigonometric values?

This is the reference table every GCSE student must know. Practise covering it and recalling each entry until the responses become automatic.

Memory trick for sin: Write the values as square roots over 2, with the numbers under the root sign counting from 0 to 4:

sin 0° = √0/2 = 0 · · · sin 30° = √1/2 = 1/2 · · · sin 45° = √2/2 · · · sin 60° = √3/2 · · · sin 90° = √4/2 = 1

The cosine column is the sine column read in reverse (from 90° back to 0°), because cos θ = sin(90° − θ).

For tan, you can always derive each value as sin ÷ cos, so you never need to memorise it separately — though knowing tan 30° = √3/3, tan 45° = 1, and tan 60° = √3 by heart will save precious minutes in the exam.

How do I use exact values in calculations?

When applying exact values, treat surds as algebraic expressions — keep them in fraction form, cancel common factors, and leave the answer as a surd unless the question says otherwise.

Worked example 1: Find the exact value of sin 30° × cos 60°.

sin 30° = 1/2 and cos 60° = 1/2

Answer: (1/2) × (1/2) = 1/4

Worked example 2: Find the exact value of tan 45° + sin 90°.

tan 45° = 1 and sin 90° = 1

Answer: 1 + 1 = 2

Worked example 3: A right-angled triangle has a hypotenuse of 10 cm and one angle of 60°. Find the exact length of the side opposite the 60° angle.

sin 60° = opposite ÷ hypotenuse, so opposite = 10 × sin 60° = 10 × (√3/2) = 5√3 cm

Worked example 4: Show that sin²30° + cos²30° = 1.

sin 30° = 1/2, so sin²30° = (1/2)² = 1/4 cos 30° = √3/2, so cos²30° = (√3/2)² = 3/4

sin²30° + cos²30° = 1/4 + 3/4 = 1 ✓

This confirms the Pythagorean identity sin²θ + cos²θ = 1 for θ = 30°. The same check works for every angle in the table.

How do I rationalise the denominator?

When an exact trigonometric value appears as a fraction with a surd in the denominator (such as 1/√2 or 1/√3), GCSE mark schemes expect you to write it in rationalised form — a denominator that contains no surd.

The method: multiply the numerator and denominator by the surd in the denominator.

Example — tan 30°: tan 30° = 1/√3

Multiply top and bottom by √3: (1 × √3) ÷ (√3 × √3) = √3 ÷ 3 = √3/3

Both 1/√3 and √3/3 are mathematically equivalent, but √3/3 is the rationalised, preferred form.

Example — sin 45°: sin 45° = 1/√2

Multiply top and bottom by √2: (1 × √2) ÷ (√2 × √2) = √2 ÷ 2 = √2/2

Quick check: (√2/2)² = 2/4 = 1/2, and sin²45° should equal 1/2 — correct. ✓

Frequently Asked Questions

Why is tan 90° undefined?

tan θ = sin θ ÷ cos θ. At 90°, cos 90° = 0, and division by zero is undefined in mathematics — the result does not exist as a finite number. On a graph of y = tan x, the curve rises steeply towards positive infinity as x approaches 90° from below, which confirms there is no single real value to assign. Writing "tan 90° = a very large number" or "∞" is incorrect; the right answer is simply "undefined".

How do I remember which of sin 30° and sin 60° is larger?

In a right-angled triangle, a larger acute angle means the opposite side is longer relative to the hypotenuse, so sine increases as the angle grows from 0° to 90°. This tells you directly that sin 60° (= √3/2 ≈ 0.866) is larger than sin 30° (= 1/2 = 0.5). By the same logic, cosine decreases as the angle grows, so cos 30° (= √3/2) is larger than cos 60° (= 1/2). The memory phrase "sin goes up, cos comes down" captures this neatly.

Do I need to memorise the derivations for the exam?

For most GCSE specifications, you are required to recall the values from the table, not reproduce the derivations. However, understanding the equilateral-triangle derivation (for 30° and 60°) and the isosceles-right-triangle derivation (for 45°) means you can reconstruct any value if memory fails under pressure — which is far more reliable than pure rote learning. Some questions also ask you to use a given triangle to prove that a specific value is correct.

What is the difference between sin²θ and sin(θ²)?

They are completely different expressions. sin²θ means (sin θ)² — you take the sine of the angle first, then square the result. For example, sin²30° = (sin 30°)² = (1/2)² = 1/4. By contrast, sin(θ²) would mean squaring the angle first, then taking the sine — for instance, sin(30²) = sin(900°), which is a different calculation entirely. The notation sin²θ is standard throughout GCSE and A-level mathematics.

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