Pythagoras' Theorem and TrigonometryEdexcel GCSE Maths: Revision notes
Section 1
How do you apply Pythagoras' theorem in 2D and 3D?
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:
- Identify the right-angled triangle within the problem
- Label the two shorter sides and the hypotenuse
- Rearrange the formula if needed: a² = c² - b² (to find a shorter side)
- Substitute values and solve
In 3D problems:
- Break down 3D shapes (cubes, cuboids, pyramids, prisms) into 2D right-angled triangles
- You may need to apply Pythagoras' theorem twice: once to find a diagonal in a 2D face, then again using that result as part of another triangle
- Common 3D scenarios include finding space diagonals in cuboids or distances from a vertex to a point on the opposite face
Key tip: Always draw or sketch the right-angled triangle clearly, marking the right angle and labelling sides correctly.
A cuboid has dimensions 3 cm, 4 cm, and 5 cm. Find the space diagonal. Step 1: Find the diagonal of the 3 cm × 4 cm base using a² + b² = c²: 3² + 4² = 9 + 16 = 25, so diagonal = 5 cm. Step 2: This diagonal, the height (5 cm), and the space diagonal form a right-angled triangle: 5² + 5² = 50, so space diagonal = √50 = 5√2 cm.
When solving 3D problems, examiners expect to see clear identification of each right-angled triangle used. Label intermediate results clearly so your working can be followed.
Section 2
What are trigonometric ratios and how do you use them?
Trigonometric ratios relate the sides and angles of a right-angled triangle. For any acute angle in a right-angled triangle:
| Ratio | Formula | Mnemonic |
|---|---|---|
| sine (sin) | opposite ÷ hypotenuse | SOH |
| cosine (cos) | adjacent ÷ hypotenuse | CAH |
| tangent (tan) | opposite ÷ adjacent | TOA |
Remember SOHCAHTOA to recall all three.
To find a missing length:
- Identify the angle you know and label the sides relative to it (opposite, adjacent, hypotenuse)
- Choose the ratio containing the known angle and the unknown side
- Rearrange if needed and solve
To find a missing angle:
- Identify which sides you know
- Choose the appropriate ratio
- Use the inverse function: sin⁻¹, cos⁻¹, or tan⁻¹ on your calculator
- Ensure your calculator is in degree mode
Common error: Confusing which side is opposite and which is adjacent—always measure relative to the angle in question, not the right angle.
In a right-angled triangle, the hypotenuse is 10 cm and one angle is 35°. Find the opposite side. Use sin(35°) = opposite ÷ 10. Rearrange: opposite = 10 × sin(35°) = 10 × 0.574 = 5.74 cm.
Students often enter angles in radian mode instead of degree mode, producing completely wrong answers. Always check your calculator display shows 'DEG' before starting trigonometry questions.
What are the exact trigonometric values you must know?
You must memorise the exact values for 0°, 30°, 45°, 60°, and 90°. These appear frequently in exam questions, and using them shows mathematical understanding rather than calculator dependence.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 or √3/3 |
| 45° | √2/2 or 1/√2 | √2/2 or 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Memory aids:
- For 30°, 45°, 60°: sine values are 1/2, √2/2, √3/2 (increasing)
- For 30°, 45°, 60°: cosine values are √3/2, √2/2, 1/2 (decreasing)
- tan = sin ÷ cos
Use these values to:
- Solve equations involving these angles without a calculator
- Leave answers in exact form (surds) when questions ask for it
- Verify calculator answers or work with special triangles
When a question says 'leave your answer in exact form' or 'in terms of surds', it signals that you should use exact values, not calculator decimals. Examiners award marks for showing this knowledge.
Learning exact values is like learning your times tables—they are the building blocks that make more complex problems faster and show mathematical fluency.
Section 4
How do you apply the sine rule and cosine rule to non-right-angled triangles?
For triangles without a right angle, standard trigonometric ratios do not apply directly. Use the sine rule or cosine rule instead.
The Sine Rule:
a/sin(A) = b/sin(B) = c/sin(C)
where lowercase letters are sides and uppercase letters are the angles opposite those sides.
Use the sine rule to find:
- A missing side when you know an angle opposite to it and another side and its opposite angle
- A missing angle when you know a side and two other angles (or a side opposite to one angle and another angle)
The Cosine Rule:
a² = b² + c² - 2bc·cos(A)
Or, rearranged to find an angle:
cos(A) = (b² + c² - a²) / (2bc)
Use the cosine rule to find:
- A missing side when you know two other sides and the included angle (the angle between them)
- A missing angle when you know all three sides
Key distinction:
- Sine rule: Use when you have an angle and its opposite side
- Cosine rule: Use when you have two sides and the included angle, or all three sides
Common setup: Label the triangle clearly with sides a, b, c and opposite angles A, B, C.
A triangle has sides 5 cm and 7 cm with an included angle of 50°. Find the third side. Use cosine rule: a² = 5² + 7² - 2(5)(7)cos(50°) = 25 + 49 - 70(0.643) = 74 - 45.01 = 28.99, so a ≈ 5.39 cm.
A triangle has sides 8 cm, 6 cm, and an angle of 40° opposite the 8 cm side. Find another angle. Use sine rule: 8/sin(40°) = 6/sin(B), so sin(B) = 6·sin(40°)/8 = 6(0.643)/8 ≈ 0.482, giving B ≈ 28.9°.
Section 5
How do you calculate triangle areas and solve 3D trigonometry problems?
Area of a triangle using trigonometry:
Area = ½ab·sin(C)
where a and b are two sides and C is the included angle between them.
When to use this formula:
- When you know two sides and the angle between them
- As an alternative to ½ × base × height when the height is not directly given
- For non-right-angled triangles where you cannot easily find the perpendicular height
3D trigonometry problems:
These typically involve finding:
- Angles between a line and a plane
- Distances in 3D space using combinations of Pythagoras' theorem and trigonometry
- Angles in pyramids, cuboids, or other 3D solids
The angle between a line and a plane:
- Identify the line and the plane
- Draw a perpendicular from the point where the line meets the plane to find the vertical height
- Identify the projection of the line onto the plane (the horizontal distance)
- Use trigonometry (usually tan) with the right-angled triangle formed by the height, projection, and the line itself
Strategy for 3D problems:
- Break 3D problems into 2D right-angled triangles
- Clearly label intermediate distances and angles
- Use Pythagoras' theorem first to find distances in the base or faces
- Then apply trigonometry to find angles or final lengths
A triangle has sides 6 cm and 8 cm with an included angle of 45°. Find the area. Area = ½(6)(8)sin(45°) = 24 × (√2/2) = 12√2 ≈ 16.97 cm².
In a pyramid with a square base of side 4 cm and height 5 cm, find the angle between a slant edge and the base plane. The horizontal distance from the base centre to a vertex is 4/√2 = 2√2 cm. Use tan(angle) = 5/(2√2) ≈ 1.768, so angle ≈ 60.3°.
For 3D angle questions, examiners want to see a clear right-angled triangle identified with the angle marked. Draw a separate 2D diagram of this triangle to show your working clearly.
Must Know
- Pythagoras' theorem (a² + b² = c²) applies to right-angled triangles in 2D and 3D; in 3D, break the problem into two right-angled triangles using intermediate distances.
- SOHCAHTOA (sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent) gives the three basic trigonometric ratios for finding missing sides or angles in right-angled triangles.
- Exact values must be memorised for 0°, 30°, 45°, 60°, and 90°; use these instead of decimals when answers are required in exact form.
- Sine rule (a/sin A = b/sin B = c/sin C) is used when you have an angle and its opposite side; cosine rule (a² = b² + c² − 2bc·cos A) is used with two sides and the included angle.
- Area = ½ab·sin(C) finds triangle area from two sides and the included angle; this formula avoids needing the perpendicular height.
- 3D trigonometry: Always identify the right-angled triangle containing the angle or distance required, often using Pythagoras first to find an intermediate length, then trigonometry to find the final answer.
That's the notes covered.
Carry on to the next subtopic.