Trigonometry: SOH CAH TOA & Sine Rule
Trigonometry links the interior angles of a triangle to the lengths of its sides. In GCSE Maths, right-angled triangles are solved using Pythagoras' theorem and the sine, cosine, and tangent ratios (SOH CAH TOA), while non-right-angled triangles on Higher tier require the Sine Rule, the Cosine Rule, and Area = ½ab sin C.
- Hypotenuse
- The longest side of a right-angled triangle, directly opposite the right angle.
- Opposite
- The side across from the angle you are working with.
- Adjacent
- The side next to the angle you are working with that is not the hypotenuse.
Right-Angled Triangle Explorer & SOH CAH TOA Formula Triangles
Drag the angle and hypotenuse sliders to see how Opposite, Adjacent, and Hypotenuse scale, or click any letter inside the formula triangle to rearrange the equation.
SOH: sin(θ) = Opposite ÷ Hypotenuse
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | - |
Exam notes: SOH CAH TOA, Exact Trig Values, Sine Rule & Cosine Rule
- Always label the three sides of a right-angled triangle relative to the given angle θ first: the Hypotenuse (H) is longest and opposite the 90° right angle; the Opposite (O) is directly across from θ; the Adjacent (A) is next to θ and touches the 90° angle.
- Use SOH CAH TOA only for right-angled triangles: sin(θ) = Opposite / Hypotenuse (SOH rule), cos(θ) = Adjacent / Hypotenuse (CAH rule), tan(θ) = Opposite / Adjacent (TOA rule). To find a missing angle, always use the inverse function: sin⁻¹(O/H), cos⁻¹(A/H), or tan⁻¹(O/A).
- For non-right-angled triangles, choose between the Sine Rule (a/sin A = b/sin B) when you have a matching opposite side-and-angle pair, and the Cosine Rule (a² = b² + c² − 2bc cos A) when you have 2 sides and the included angle (SAS) or all 3 sides (SSS).
- On Non-Calculator Paper 1, you must memorise the exact trig values for 0°, 30°, 45°, 60°, and 90°: sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3; sin(45°) = cos(45°) = √2/2, tan(45°) = 1; sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3.
| Rule / Formula | When to Use It | Exact Rearrangement |
|---|---|---|
| SOH Rule: sin(θ) = O ÷ H | Right-angled triangle involving Opposite & Hypotenuse | O = H × sin(θ) | H = O ÷ sin(θ) | θ = sin⁻¹(O ÷ H) |
| CAH Rule: cos(θ) = A ÷ H | Right-angled triangle involving Adjacent & Hypotenuse | A = H × cos(θ) | H = A ÷ cos(θ) | θ = cos⁻¹(A ÷ H) |
| TOA Rule: tan(θ) = O ÷ A | Right-angled triangle involving Opposite & Adjacent | O = A × tan(θ) | A = O ÷ tan(θ) | θ = tan⁻¹(O ÷ A) |
| Sine Rule: a/sin A = b/sin B | Any triangle with a known opposite side-angle pair | Find side: a = b sin A / sin B | Find angle: sin A = a sin B / b |
| Cosine Rule: a² = b² + c² − 2bc cos A | Any triangle given SAS (2 sides + trapped angle) or SSS | Find angle: cos A = (b² + c² − a²) ÷ (2bc) |
| Area Rule: Area = ½ab sin C | Area of any non-right triangle given 2 sides + trapped angle | Multiply ½ × side a × side b × sin(included angle C) |
Key Concepts & Exam Definitions (8)
- Hypotenuse
- The longest side of a right-angled triangle, directly opposite the right angle.
- Opposite
- The side across from the angle you are working with.
- Adjacent
- The side next to the angle you are working with that is not the hypotenuse.
- Sine Rule
- The ratio of a side to the sine of its opposite angle is constant for all three sides of a triangle: a/sin(A) = b/sin(B) = c/sin(C).
- Cosine Rule
- An extension of Pythagoras theorem for non-right-angled triangles: a² = b² + c² - 2bc cos(A).
- Exact Trigonometric Values (30°, 45°, 60°)
- Derived from an equilateral triangle of side 2 split in half (giving angles 30° and 60° with height √3) and an isosceles right-angled triangle of legs 1 and 1 (giving hypotenuse √2 and angle 45°).
- Angles of Elevation and Depression
- Both are measured from the horizontal line. The angle of elevation (looking up) equals the angle of depression (looking down) because they are alternate angles ('Z-angles') between parallel horizontal lines.
- 3D Trigonometry & Pythagoras
- To find the angle between a space diagonal and the base of a cuboid, first use Pythagoras on the base rectangle to find the base diagonal d = √(l² + w²), then use tan(θ) = height ÷ d in the vertical right-angled triangle.
Quick-Fire Flashcards (8)
Click any card to check the answer, then mark it as Known to track your progress.
What is the formula for calculating tan(θ)?
What is the exact value of sin(30°)?
Which trigonometric rule should you use if you know two sides and the included angle?
What is the exact value of cos(60°)?
What is the formula for the area of any triangle using trigonometry?
State the SOH, CAH, and TOA rules in full.
What are the exact values of sin(30°), cos(60°), and tan(45°) without a calculator?
How do you decide whether to use the Sine Rule or the Cosine Rule on a non-right-angled triangle?
Exam-Style Questions & Mark Schemes (5)
Write or think through your answer first, then click Check Mark Scheme to see the exact points examiners award marks for.
- Q1
A right-angled triangle has a hypotenuse of 10cm and an angle of 30°. Find the length of the side opposite the 30° angle.
Click “Check Mark Scheme” to compare your answer, 5cm. sin(30°) = Opp / 10, so 0.5 * 10 = 5.
- Q2
In triangle ABC, angle A = 40°, side a = 8cm, and side b = 10cm. Set up the equation using the Sine Rule to find angle B.
Click “Check Mark Scheme” to compare your answer, sin(B) / 10 = sin(40°) / 8
- Q3
Find the exact value of tan(45°).
Click “Check Mark Scheme” to compare your answer, 1
- Q4
In a right-angled triangle, the hypotenuse is 14 cm and the angle θ is 38°. Calculate the length of the side adjacent to the 38° angle to 1 decimal place. [2 marks]
Click “Check Mark Scheme” to compare your answer, 1) We have Adjacent (A) and Hypotenuse (H = 14 cm), so use CAH: cos(38°) = A ÷ 14 [1 mark]. 2) Rearrange: A = 14 × cos(38°) = 14 × 0.7880 = 11.0 cm (to 1 d.p) [1 mark].
- Q5
In triangle ABC, AB = 8 cm, AC = 11 cm, and angle BAC = 65°. Calculate the length of side BC and the area of triangle ABC to 3 significant figures. [5 marks]
Click “Check Mark Scheme” to compare your answer, 1) Two sides and the included angle (SAS) → use Cosine Rule: BC² = 8² + 11² − 2(8)(11)cos(65°) [1 mark]. 2) BC² = 64 + 121 − 176(0.4226) = 185 − 74.38 = 110.62 ⇒ BC = √110.62 = 10.5 cm [2 marks]. 3) Area = ½ab sin C = ½ × 8 × 11 × sin(65°) = 44 × 0.9063 = 39.9 cm² [2 marks].
Common Exam Questions (3)
What does SOH CAH TOA stand for?
SOH CAH TOA is a mnemonic for the three trigonometric ratios in a right-angled triangle: sin(θ) = Opposite ÷ Hypotenuse, cos(θ) = Adjacent ÷ Hypotenuse, and tan(θ) = Opposite ÷ Adjacent.
When do I use the Sine Rule?
Use the Sine Rule, a/sin(A) = b/sin(B), in a non-right-angled triangle when you know a matching pair of a side length and its opposite angle, plus one other side or angle.
Do I need to memorise exact trigonometric values?
Yes. On the non-calculator GCSE paper, you are expected to recall the exact values of sin, cos, and tan for 0°, 30°, 45°, 60°, and 90°.
Need more questions on Trigonometry: SOH CAH TOA & Sine Rule, or want to organise your own notes?
Type a specific exam question you are stuck on, pick one of the prompts below, or paste your class notes to build a custom set of flashcards and practice questions on this page.
Related study guides
Geometry: Angles, Shapes & Proofs
Geometry studies the properties of angles, lines, polygons, and circles. Securing marks in GCSE geometry relies on stating exact geometric reasons alongside your calculations, including parallel-line angle facts, interior and exterior polygon formulas, circle theorems, and triangle congruence criteria.
Quadratic Equations & The Quadratic Formula
A quadratic expression takes the form ax² + bx + c (where a ≠ 0) and plots as a symmetrical parabola. In GCSE Maths, you solve quadratic equations by factorising into two brackets, using the quadratic formula, or completing the square to find the roots, y-intercept, and turning point.