Trigonometry and PythagorasAQA GCSE Maths: Revision notes

Pythagoras' theorem states that in a right-angled triangle, a² + b² = c², where c is the hypotenuse (the longest side opposite the right angle).

In 2D problems:

• Use the theorem directly to find missing side lengths
• Rearrange to find any side: a = √(c² - b²) or c = √(a² + b²)
• Apply to real-world contexts like distances, diagonals in rectangles, or distances between coordinates

In 3D problems:

• Break down 3D shapes into 2D right-angled triangles
• Common contexts: finding diagonals of cuboids, distances between vertices, or slant heights of pyramids
• Apply Pythagoras twice if necessary (first to find a 2D diagonal, then use that as a side in another right triangle)

Key steps for 3D problems:

1. Identify the right-angled triangle within the 3D shape
2. Label all known measurements
3. Apply Pythagoras to find the required distance
4. For compound problems, use your answer from step 3 in a subsequent calculation

Trigonometric ratios (sin, cos, tan) relate the angles and sides of right-angled triangles. Use the mnemonic SOHCAHTOA:

Finding lengths:

• Identify the angle you know
• Label the sides: opposite (across from the angle), adjacent (next to the angle), hypotenuse
• Choose the ratio that involves the angle and the sides you know/need
• Rearrange if needed: opposite = sin θ × hypotenuse, etc.

Finding angles:

• Use inverse trigonometric functions: sin⁻¹, cos⁻¹, tan⁻¹
• Example: if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°
• Always check your calculator is in degree mode, not radian mode

You must know the exact values of sin, cos, and tan for special angles. These are frequently tested and cannot be approximated using a calculator.

Key patterns to remember:

• At 0°: sine is 0, cosine is 1 (full horizontal reach)
• At 90°: sine is 1, cosine is 0 (full vertical reach)
• At 45°: both sine and cosine are equal (√2/2)
• sin and cos swap values between 30° and 60°
• Use rationalised denominators when writing answers (e.g., √3/3 rather than 1/√3)

When to use exact values:

• When a question says "exact" or "leave your answer in surd form"
• Never use a calculator; substitute the exact value directly
• Simplify surds fully in your final answer

3D trigonometry combines both trigonometric ratios and Pythagoras to solve problems in three-dimensional contexts.

Common 3D scenarios:

• Finding angles between a line and a plane (e.g., the slant height of a pyramid and the base)
• Finding distances between vertices or points in cuboids, pyramids, and prisms
• Calculating heights or diagonals using multi-step approaches

Strategy for 3D problems:

1. Identify a 2D right-angled triangle within the 3D shape (or create one by projecting a line onto a plane)
2. Use Pythagoras to find any missing 2D measurements if needed
3. Apply trigonometric ratios to find angles or additional lengths
4. Draw and label a clear 2D diagram of the triangle you're working with

Example contexts:

• Angle between a slant edge and the base of a pyramid
• Angle of elevation or depression from a point to another point in 3D space
• Distance from a vertex to a face diagonal in a cuboid

Always identify the right angle in your 2D triangle before applying ratios.

For non-right-angled (oblique) triangles, you cannot use basic trigonometric ratios. Instead, use the sine rule, cosine rule, and the area formula.

The Sine Rule:

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

• Use this to find sides when you know an angle and an opposite side, plus another angle or side
• Use this to find angles when you know two sides and an angle opposite one of them
• Ambiguous case: When finding an angle, you may get two possible answers (use context to choose)

The Cosine Rule:

a² = b² + c² − 2bc cos A

Or rearranged to find an angle:

cos A = (b² + c² − a²) / 2bc

• Use this to find a side when you know two other sides and the included angle (SAS)
• Use this to find an angle when you know all three sides (SSS)

Area Formula (non-right-angled triangle):

Area = ½ab sin C

• Where a and b are two sides and C is the included angle (between them)
• More useful than ½ × base × height when you don't know the perpendicular height

Choosing the right formula:

Bearings are directions measured clockwise from north. Bearing problems combine trigonometry and Pythagoras to find distances and directions between points.

Key bearing facts:

• Bearings are always written as three figures (e.g., 050°, 120°, 305°)
• North is 000°, East is 090°, South is 180°, West is 270°
• A bearing from A to B is the direction you face when standing at A looking towards B
• The reverse bearing (from B to A) differs from the original by 180° (if result > 360°, subtract 360°)

Solving bearing problems:

1. Sketch a clear diagram with north arrows at each point
2. Mark bearings as angles clockwise from north
3. Identify right-angled triangles using north lines as reference
4. Use Pythagoras to find distances if needed
5. Use trigonometry (often tan) to find bearing angles
6. Convert angles back to bearings (add to north direction)

Common setups:

• Two ships or walkers moving from a starting point on different bearings
• Finding the direct distance between two points given their bearings and distances from a third point
• Finding the bearing of one point from another after calculating an angle

Important: Always measure bearings from the north line, not from the previous direction. Draw north arrows to avoid confusion.