Exact Trig Values, Sine Rule, Cosine Rule & Area of Any Triangle

SPEC G21, G22, G23 GEOMETRY & MEASURES

⭐ HIGH PRIORITY F+H WITH HIGHER SECTIONS

Exact Trig Values, Sine & Cosine Rules

The exact values you must know for non-calculator papers, plus the two rules that handle every non-right triangle problem at Higher tier.

What you will learn

By the end of this lesson you will:

(F+H) Recall the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90° for use on non-calculator papers

(F+H) Derive these exact values from two key triangles: the 30-60-90 half-equilateral and the 45-45-90 right isosceles

[Higher] Apply the sine rule when you know a side–opposite-angle pair

[Higher] Apply the cosine rule for SAS (find a side) or SSS (find an angle)

[Higher] Use Area = ½ab sin C for the area of any triangle

[Higher] Choose the right rule from the information given

Critical — not on the formula sheet

For AQA 8300, sine rule, cosine rule and Area = ½ab sin C must be memorised. They appear in the "know without being given" section of the spec. The exact trig values must also be memorised — especially for Paper 1 (non-calculator).

Why we need new tools

SOH CAH TOA from Lesson 4.16 only works on right-angled triangles. For any other triangle, we need different methods:

Right-angled triangle

Use Pythagoras (4.15) for sides and SOH CAH TOA (4.16) for sides and angles.a² + b² = c²

sinθ = O/H, cosθ = A/H, tanθ = O/A

Any other triangle

Use the sine rule or cosine rule (Higher tier). For area, use ½ab sin C.a/sin A = b/sin B = c/sin C

a² = b² + c² − 2bc cos A

Area = ½ab sin C

Notation convention

In any triangle ABC, the lowercase letter labels the side opposite the angle with the matching capital letter. So side a is opposite angle A, side b is opposite angle B, side c is opposite angle C.

Exact Trig Values — Memorise This Table

On Paper 1 (non-calculator) any trig value comes from this table. The angles are 0°, 30°, 45°, 60° and 90°.

θ0°30°45°60°90°

sin θ0½2 / 23 / 21

cos θ13 / 22 / 2½0

tan θ01 / 313undefined

Pattern to help recall

Read sin θ left to right: 0/2, 1/2, 2/2, 3/2, 4/2 — or written cleanly: 0, ½, 2/2, 3/2, 1. Cos values are the reverse (sin and cos are mirror images either side of 45°). Tan = sin / cos.

Most-confused value

cos 60° = ½ (not 3/2). Examiner reports consistently identify cos 60° as the value students mis-remember most often. Memorise it as the mirror of sin 30°.

Where the Exact Values Come From

You don't have to memorise the table cold — every value comes from two simple triangles:

From 30-60-90

sin 30° = opp/hyp = 1/2. cos 30° = adj/hyp = 3/2. tan 30° = opp/adj = 1/ 3.

Swap perspective for 60°: sin 60° = 3/2, cos 60° = 1/2, tan 60° = 3.

From 45-45-90

Both non-right angles are 45°. sin 45° = cos 45° = 1/ 2 = 2/2. tan 45° = 1.

For 0° and 90° the values come from limits: sin 0° = 0, cos 0° = 1, tan 0° = 0. sin 90° = 1, cos 90° = 0, tan 90° is undefined (would divide by 0).

Worked Example — Exact Values01

Show that tan 60° × sin 60° = 3/2.

You must show all your working. Do not use a calculator. [3 marks]

Substitute the exact values: tan 60° = 3 and sin 60° = 3/2. Multiply, then simplify.

C — Calculate

tan 60° × sin 60° = 3 × 3/2

= ( 3 × 3) / 2 = 3/2

E — Evaluate

tan 60° × sin 60° = 3/2 ✓

On a "show that" question, every step has to appear — substitution, simplification, final equality. Skipping a line typically loses an M-mark even if the final value is correct.

Sine Rule ⬣ HIGHER

For any triangle ABC (right-angled or not):

a / sin A = b / sin B = c / sin C

⬣ Higher — When to use sine rule

Sine rule needs a matched side–angle pair (one side and the angle directly opposite to it). Use it when:

AAS / ASA — two angles and any one side, find another side

SSA — two sides and an angle opposite one of them, find the other angle

Tip: when finding a side, write the rule with sides on top: a/sinA = b/sinB. When finding an angle, flip it: sinA/a = sinB/b.

SSA ambiguous case

For SSA configurations, sin⁻ can give two valid answers — the acute angle θ or the obtuse angle (180° − θ). Both may produce a real triangle. Read the diagram or context: if the question or sketch shows an obtuse angle, take 180° − θ instead of the calculator's acute value.

Worked Example — Sine Rule

⬣ HIGHER

In triangle PQR, angle P = 30°, angle Q = 45°, and side p = 10 cm. Find the length of side q. Give your answer in exact form. [3 marks]

M — Method

I have a side–angle pair (p, P) and another angle (Q) — sine rule applies. To find a side, place sides on top:

p / sin P = q / sin Q → q = p sin Q / sin P

C — Calculate

q = 10 sin 45° / sin 30°

Substituting exact values: sin 45° = 2/2 and sin 30° = 1/2.

q = 10 ( 2/2) / (1/2) = 10 ( 2/2) 2 = 10 2 cm

E — Evaluate

q = 10 2 cm ≈ 14.1 cm (3 s.f.)

When the question asks for an "exact form" answer, leave it as a surd: 10 2 cm. The decimal 14.1 is the calculator approximation.

Cosine Rule ⬣ HIGHER

For any triangle — an extension of Pythagoras with a correction term for non-right angles:

a² = b² + c² − 2bc cos A

where A is the angle between sides b and c, and a is the side opposite A.

⬣ Higher — When to use cosine rule

SAS (two sides + the angle between them) → find the third side. Use the formula directly.

SSS (all three sides) → find any angle. Rearrange to cos A = (b² + c² − a²) / (2bc) .

Sign trap

The cosine rule has a minus sign: a² = b² + c² − 2bc cos A. Writing "+" instead of "−" is a top examiner-flagged error and produces nonsense (typically a third side longer than the other two combined).

Worked Example — Cosine Rule (SAS)

⬣ HIGHER

In triangle ABC, AB = 8 cm, AC = 5 cm, and angle BAC = 60°. Calculate the length of BC. [3 marks]

M — Method

Two sides (AB, AC) and the included angle (BAC). This is SAS — use the cosine rule directly. Setting a = BC (the unknown opposite A), b = AC = 5, c = AB = 8:

BC² = AB² + AC² − 2 AB AC cos(BAC)

C — Calculate

BC² = 8² + 5² − 2(8)(5)cos 60°

Use cos 60° = ½:

BC² = 64 + 25 − 80 ½ = 89 − 40 = 49

BC = 49 = 7 cm

E — Evaluate

BC = 7 cm

The 5-7-8 triangle with a 60° angle (opposite the 7) is a clean exam favourite — the cos 60° = ½ collapses the formula to integer arithmetic.

Cosine Rule for an Angle ⬣ HIGHER

When you know all three sides (SSS) and want an angle, rearrange the cosine rule to make cos A the subject:

cos A = (b² + c² − a²) / (2bc)

⬣ Higher — DerivationStart with a² = b² + c² − 2bc cos A. Add 2bc cos A to both sides, then subtract a²:2bc cos A = b² + c² − a²Divide both sides by 2bc:cos A = (b² + c² − a²) / (2bc)Then take cos⁻ on both sides to get A.

Sign telling you the angle's type

If cos A is positive, A is acute (less than 90°).

If cos A is zero, A is exactly 90° (this is a right triangle — Pythagoras emerges).

If cos A is negative, A is obtuse (greater than 90°).

Worked Example — Cosine Rule (SSS)

⬣ HIGHER

In triangle ABC, side a = 13 cm, side b = 7 cm, side c = 8 cm. Calculate the size of angle A. [3 marks]

M — Method

All three sides given (SSS) — use the rearranged cosine rule:

cos A = (b² + c² − a²) / (2bc)

C — Calculate

cos A = (7² + 8² − 13²) / (2 7 8)

= (49 + 64 − 169) / 112 = −56 / 112 = −½

A = cos⁻ (−½) = 120°

E — Evaluate

A = 120°

The negative value of cos A confirms A is obtuse, which matches the diagram. The largest angle is always opposite the longest side — here side a = 13 is the longest, so angle A is the largest.

Area of Any Triangle ⬣ HIGHER

For triangles where you don't have a perpendicular height, the area can be calculated from two sides and the angle between them:

Area = ½ab sin C

where a and b are two sides, and C is the angle between them.

⬣ Higher — Where this formula comes from

The standard area formula is ½ × base × perpendicular height. Drop a perpendicular from one vertex; its length equals (other side) × sin(included angle). Substituting gives Area = ½ab sin C.

Use the included angle

The angle you use must be the one between the two named sides — not any other angle. Using the wrong angle is a top-marked error.

Worked Example — Area

⬣ HIGHER

In triangle ABC, AB = 6 cm, AC = 8 cm, and angle BAC = 30°. Calculate the area of the triangle. Do not use a calculator. [3 marks]

I have two sides (AB, AC) meeting at A with the angle between them. Use Area = ½ × AB × AC × sin(BAC).

C — Calculate

Area = ½ 6 8 sin 30°

Use sin 30° = ½:

Area = ½ 6 8 ½ = 24 ½ = 12

E — Evaluate

Area = 12 cm²

A non-calculator question relies on you recognising sin 30° = ½. The arithmetic collapses cleanly because of this.

Which Rule? — The Decision

Read the question carefully. Identify what you have, then pick the rule:

What you knowWhat to useTier

Two sides + the right angle, find the third sidePythagoras: a² + b² = c²F+H

One side + one non-right angle in a right triangleSOH CAH TOAF+H

Side–angle pair (a known side opposite a known angle) — AAS, ASA, SSASine rule : a/sinA = b/sinB⬣ Higher

Two sides + the angle between them — SASCosine rule for a side: a² = b² + c² − 2bc cos A⬣ Higher

All three sides — SSSCosine rule rearranged for an angle: cos A = (b² + c² − a²) / (2bc)⬣ Higher

Area, two sides + included angleArea = ½ab sin C⬣ Higher

Test before you start

Look for a matched side–angle pair first. If you find one, sine rule is usually the fastest route. If not, you almost certainly need the cosine rule.

Examiner Alerts

Alert 1 — Decimals where exact form is required

On Paper 1 or any "exact form" question, students compute sin 60° on a calculator (gets 0.866) and lose the accuracy mark. What to write instead: use the exact value 3/2 throughout. The final answer should contain a surd or a fraction — never a rounded decimal.

Alert 2 — Cosine rule sign error

The formula has a minus: a² = b² + c² − 2bc cos A. Writing "+" gives a third side longer than the sum of the other two — geometrically impossible. What to write instead: always write the formula with the minus sign before substituting numbers; double-check the sign before evaluating cos A.

Alert 3 — Mishandling fractions in "show that" questions

When multiplying surds and fractions, errors are common: 2 × 3/2 = 3 (not 6/2 or 6/4). What to write instead: show every step explicitly — cancel the 2 in the numerator with the 2 in the denominator before computing anything else.

Common Errors

Students often writeCorrect versionWhy it matters

cos 60° = 3/2cos 60° = ½Most-confused exact value — mirror of sin 30°.

a² = b² + c² + 2bc cos Aa² = b² + c² − 2bc cos AThe minus is essential — with plus, the formula is wrong.

Sine rule for SAS dataSAS → cosine rule (sine rule needs a matched pair)Sine rule fails without an opposite angle.

Area = ½ab sin (any angle)Area = ½ab sin C, where C is between sides a and bThe angle must be the included angle.

2 × 3/2 = 6/22 × 3/2 = 3Multiplying numerator by 2 cancels the 2 in the denominator. Surds are not multiplied into integers.

Area = ½ base × slantArea = ½ base × perpendicular height — or use ½ab sin CSlant length is not the perpendicular height of a triangle.

In SSA: take only the calculator's acute answerCheck whether 180° − (calc value) gives a valid triangleThe sine function gives the same value at θ and 180°−θ. The diagram or question context tells you which to keep.

Practice ⬣ HIGHER

In triangle ABC, AB = 5 cm, BC = 7 cm, AC = 8 cm.

(a) Calculate the size of angle BAC. [3 marks]

(b) Hence calculate the area of the triangle. Give your answer in exact form. [2 marks]

M — Method

(a) All three sides given (SSS). Use cos A = (b² + c² − a²) / (2bc), with a = BC = 7 (opposite A), b = AC = 8, c = AB = 5.

(b) Use Area = ½bc sin A with the angle just found and the two sides meeting at A.

C — Calculate

(a)

cos A = (8² + 5² − 7²) / (2 8 5)

= (64 + 25 − 49) / 80 = 40 / 80 = ½

A = cos⁻ (½) = 60°

(b)

Area = ½ 8 5 sin 60° = 20 3/2 = 10 3

E — Evaluate

(a) Angle BAC = 60° (b) Area = 10 3 cm²

10 3 ≈ 17.3 cm² on a calculator. The 5-7-8 triangle is a famous configuration: the angle opposite the side of length 7 is exactly 60°, and the area in exact form is 10 3.

Lesson checkpoint — you should now be able to:

(F+H) Recall and apply exact values for sin, cos, tan at 0°, 30°, 45°, 60°, 90°

[Higher] Apply the sine rule to find a missing side or angle when there's a matched side–angle pair

[Higher] Apply the cosine rule for SAS (find a side) and SSS (find an angle)

[Higher] Use Area = ½ab sin C for the area of any triangle

[Higher] Choose the right rule from the information given in the question