Measurement, Space & Statistics

Pythagoras' Theorem and Right-Angled Trigonometry

Year 9

✓Apply Pythagoras' theorem to find unknown side lengths in right-angled triangles.
✓Identify the hypotenuse, opposite, and adjacent sides relative to a given angle in a right-angled triangle.
✓Define and apply the sine, cosine, and tangent ratios to solve problems involving right-angled triangles.
✓Calculate unknown side lengths using trigonometric ratios.
✓Calculate unknown angle sizes using inverse trigonometric ratios.

Key concepts

Pythagoras' Theorem

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. This theorem is fundamental for finding unknown side lengths in right-angled triangles.

a^2 + b^2 = c^2 (where 'c' is the hypotenuse)

Right-Angled Triangle Terminology

To apply trigonometry, we need to correctly identify the sides of a right-angled triangle relative to a specific reference angle (θ):\n- Hypotenuse: The longest side, always opposite the right angle.\n- Opposite: The side directly across from the reference angle (θ).\n- Adjacent: The side next to the reference angle (θ) that is not the hypotenuse.

Trigonometric Ratios (SOH CAH TOA)

These ratios relate the angles of a right-angled triangle to the lengths of its sides. The mnemonic SOH CAH TOA is used to remember them:\n- Sine (SOH): Sine of an angle is the ratio of the length of the Opposite side to the length of the Hypotenuse.\n- Cosine (CAH): Cosine of an angle is the ratio of the length of the Adjacent side to the length of the Hypotenuse.\n- Tangent (TOA): Tangent of an angle is the ratio of the length of the Opposite side to the length of the Adjacent side.

sin(θ) = Opposite / Hypotenuse\ncos(θ) = Adjacent / Hypotenuse\ntan(θ) = Opposite / Adjacent

Inverse Trigonometric Ratios

When you know the lengths of two sides of a right-angled triangle and need to find the size of an unknown angle, you use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹). These functions 'undo' the regular trigonometric functions.

θ = sin⁻¹(Opposite / Hypotenuse)\nθ = cos⁻¹(Adjacent / Hypotenuse)\nθ = tan⁻¹(Opposite / Adjacent)

Key facts to remember

1Pythagoras' theorem (a^2 + b^2 = c^2) applies exclusively to right-angled triangles, where 'c' is always the hypotenuse.
2The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
3The mnemonic SOH CAH TOA helps remember the trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
4Use inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to calculate the size of an unknown angle when two side lengths are known.
5Always ensure your calculator is set to DEGREE mode when performing trigonometric calculations involving angles in degrees.
6Always include appropriate units (e.g., cm, m, °) in your final answers.

Worked examples

Example 1

A right-angled triangle has two shorter sides measuring 8 cm and 15 cm. Calculate the length of the hypotenuse.

IIdentify the knowns: a = 8 cm, b = 15 cm. We need to find c (hypotenuse).

IIApply Pythagoras' theorem: a^2 + b^2 = c^2

IIISubstitute the values: 8^2 + 15^2 = c^2

IVCalculate the squares: 64 + 225 = c^2

VAdd the values: 289 = c^2

VITake the square root of both sides: c = sqrt(289)

VIICalculate the final value: c = 17

Answer

The length of the hypotenuse is 17 cm.

Remember that 'c' always represents the hypotenuse.

Example 2

A ladder leans against a wall, making an angle of 65° with the ground. If the base of the ladder is 1.5 m from the wall, how high up the wall does the ladder reach? (Round your answer to two decimal places).

IDraw a diagram and label the knowns: angle θ = 65°, Adjacent side = 1.5 m. We need to find the Opposite side (height, h).

IIIdentify the relevant trigonometric ratio: We have the Adjacent side and need the Opposite side, so we use TOA (Tangent).

IIIWrite the formula: tan(θ) = Opposite / Adjacent

IVSubstitute the known values: tan(65°) = h / 1.5

VRearrange to solve for h: h = 1.5 * tan(65°)

VICalculate the value (ensure calculator is in DEGREE mode): h ≈ 1.5 * 2.1445

VIIRound to two decimal places: h ≈ 3.22

Answer

The ladder reaches approximately 3.22 m up the wall.

Always check your calculator is in DEGREE mode for these calculations.

Example 3

A ramp is 6 m long and rises vertically by 2 m. Calculate the angle of elevation of the ramp to the nearest degree.

IDraw a diagram and label the knowns: Hypotenuse = 6 m, Opposite side = 2 m. We need to find the angle θ.

IIIdentify the relevant trigonometric ratio: We have the Opposite side and the Hypotenuse, so we use SOH (Sine).

IIIWrite the formula: sin(θ) = Opposite / Hypotenuse

IVSubstitute the known values: sin(θ) = 2 / 6

VSimplify the ratio: sin(θ) = 1 / 3

VIUse the inverse sine function to find θ: θ = sin⁻¹(1/3)

VIICalculate the value (ensure calculator is in DEGREE mode): θ ≈ 19.47°

VIIIRound to the nearest degree: θ ≈ 19°

Answer

The angle of elevation of the ramp is approximately 19°.

Use sin⁻¹, cos⁻¹, or tan⁻¹ when finding an unknown angle.

Common mistakes

✗Incorrectly identifying the hypotenuse, opposite, or adjacent sides relative to the given angle.
✗Attempting to use Pythagoras' theorem or trigonometric ratios on triangles that are not right-angled.
✗Mixing up the trigonometric ratios (e.g., using cos when sin is required) or applying them incorrectly.
✗Forgetting to use the inverse trigonometric functions (e.g., sin⁻¹) when calculating an unknown angle.
✗Having the calculator in the wrong mode (e.g., radians instead of degrees), leading to incorrect angle calculations.
✗Rounding intermediate steps during calculations, which can lead to inaccuracies in the final answer.

Exam tips

★Always draw a clear diagram for each problem and label all known and unknown sides and angles. This helps in visualising the problem.
★Clearly identify whether you need to use Pythagoras' theorem (for sides only) or trigonometry (for sides or angles).
★Show all your working steps, including the formula used, substitution of values, and the calculation. This allows for partial marks even if the final answer is incorrect.
★Check the reasonableness of your answer. For example, the hypotenuse must always be the longest side, and angles in a triangle must sum to 180°.

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