Geometry & measures

Pythagoras' Theorem and Basic Trigonometry

Year 10 · Year 11

✓Apply Pythagoras' Theorem to calculate unknown side lengths in two-dimensional right-angled triangles.
✓Extend the application of Pythagoras' Theorem to solve problems involving lengths in three-dimensional shapes.
✓Use the trigonometric ratios (sine, cosine, tangent) to find unknown side lengths and angles in right-angled triangles (SOHCAHTOA).
✓Recall and accurately use the exact trigonometric values for 0°, 30°, 45°, 60°, and 90°.

Key concepts

Pythagoras' Theorem (2D)

Pythagoras' Theorem states that in any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs). This theorem is fundamental for finding unknown side lengths when two sides are known.

a² + b² = c² (where 'c' is the hypotenuse and 'a' and 'b' are the other two sides)

Pythagoras' Theorem (3D)

To apply Pythagoras' Theorem in three dimensions, you typically need to use it twice. First, find a diagonal on one of the faces of the 3D shape, creating a 2D right-angled triangle. Then, use this newly found length as one of the sides in a second right-angled triangle to find the final unknown length, often a space diagonal.

SOHCAHTOA (Trigonometric Ratios)

SOHCAHTOA is a mnemonic used to remember the three primary trigonometric ratios for right-angled triangles. These ratios relate the angles of a right-angled triangle to the lengths of its sides. * SOH: Sine = Opposite / Hypotenuse * CAH: Cosine = Adjacent / Hypotenuse * TOA: Tangent = Opposite / Adjacent The 'opposite' side is across from the angle you are considering. The 'adjacent' side is next to the angle, not the hypotenuse. The 'hypotenuse' is always the longest side, opposite the right angle.

sin(θ) = Opposite / Hypotenuse, cos(θ) = Adjacent / Hypotenuse, tan(θ) = Opposite / Adjacent

Exact Trigonometric Values

Certain angles have trigonometric ratios that can be expressed exactly as fractions or surds, rather than decimal approximations. These 'exact values' are important for non-calculator questions and for understanding the fundamental properties of trigonometric functions. You are expected to recall these values for 0°, 30°, 45°, 60°, and 90°.

Key facts to remember

1Pythagoras' Theorem: In a right-angled triangle, a² + b² = c², where 'c' is the hypotenuse.
2SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
3The hypotenuse is always the longest side and is opposite the right angle.
4To find an angle using trigonometry, use the inverse functions: sin⁻¹, cos⁻¹, tan⁻¹.
5Exact trigonometric values for common angles: sin(0°)=0, cos(0°)=1, tan(0°)=0; sin(30°)=1/2, cos(30°)=√3/2, tan(30°)=1/√3; sin(45°)=√2/2, cos(45°)=√2/2, tan(45°)=1; sin(60°)=√3/2, cos(60°)=1/2, tan(60°)=√3; sin(90°)=1, cos(90°)=0, tan(90°) is undefined.
6When using Pythagoras in 3D, you typically apply the theorem twice.

Worked examples

Example 1

A right-angled triangle has sides of length 5 cm and 12 cm. Calculate the length of the hypotenuse.

ILet the unknown hypotenuse be 'c'. The other two sides are a = 5 cm and b = 12 cm.

IIUsing Pythagoras' Theorem: a² + b² = c²

III5² + 12² = c²

IV25 + 144 = c²

V169 = c²

VIc = √169

Answer

c = 13 cm

Example 2

In a right-angled triangle, an angle is 35° and the hypotenuse is 10 cm. Find the length of the side opposite the 35° angle, to 3 significant figures.

IIdentify the knowns and unknowns: Angle = 35°, Hypotenuse (H) = 10 cm, Opposite (O) = x (unknown).

IIChoose the correct ratio: SOH (Sine = Opposite / Hypotenuse).

IIIsin(35°) = x / 10

IVx = 10 × sin(35°)

Vx = 10 × 0.57357...

Answer

x = 5.74 cm (3 s.f.)

Ensure your calculator is in 'DEG' (degrees) mode.

Example 3

A right-angled triangle has an adjacent side of 8 cm and a hypotenuse of 11 cm. Find the angle between these two sides, to 1 decimal place.

IIdentify the knowns and unknowns: Adjacent (A) = 8 cm, Hypotenuse (H) = 11 cm, Angle (θ) = unknown.

IIChoose the correct ratio: CAH (Cosine = Adjacent / Hypotenuse).

IIIcos(θ) = 8 / 11

IVcos(θ) = 0.72727...

Vθ = cos⁻¹(0.72727...)

Answer

θ = 43.3° (1 d.p.)

Use the inverse cosine function (cos⁻¹) to find the angle.

Example 4

A cuboid has dimensions 3 cm by 4 cm by 12 cm. Find the length of the space diagonal from one corner to the opposite corner.

IStep 1: Find the diagonal of the base. Let the base dimensions be 3 cm and 4 cm. Let the base diagonal be 'd'.

IId² = 3² + 4²

IIId² = 9 + 16

IVd² = 25

Vd = √25 = 5 cm

VIStep 2: Use the base diagonal and the height to find the space diagonal. The height is 12 cm. Let the space diagonal be 'D'.

VIID² = d² + 12²

VIIID² = 5² + 12²

9D² = 25 + 144

10D² = 169

11D = √169

Answer

D = 13 cm

This involves two applications of Pythagoras' Theorem.

Example 5

Without using a calculator, evaluate sin(30°) + cos(60°) - tan(45°).

IRecall the exact values: sin(30°) = 1/2

IIcos(60°) = 1/2

IIItan(45°) = 1

IVSubstitute the values into the expression: (1/2) + (1/2) - 1

V1 - 1

Answer

Memorising these exact values is crucial for non-calculator papers.

Common mistakes

✗Incorrectly identifying the hypotenuse (it's always opposite the right angle and the longest side).
✗Mixing up the opposite and adjacent sides relative to the given angle in trigonometry problems.
✗Using the wrong trigonometric ratio (e.g., using sine instead of cosine).
✗Forgetting to take the square root as the final step when using Pythagoras' Theorem.
✗Rounding intermediate steps in calculations, leading to inaccurate final answers.
✗Not knowing or incorrectly recalling the exact trigonometric values.
✗Using a calculator in radian mode instead of degree mode for angle calculations.

Exam tips

★Always draw a clear diagram and label the sides (Opposite, Adjacent, Hypotenuse) relative to the angle you are working with.
★Write down 'SOHCAHTOA' at the start of any trigonometry question to help you choose the correct ratio.
★Show all your working steps clearly, even for simple calculations, as method marks are often awarded.
★Check if your answer makes sense: the hypotenuse should always be the longest side; angles in a triangle sum to 180°.
★For exact value questions, be prepared to work without a calculator and show the exact form (e.g., fractions or surds).

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum