Section B — Chapters 23–39

Introduction to Trigonometry: Right-angled Triangles

Junior Cycle

✓By the end of this lesson students will be able to identify the hypotenuse, opposite, and adjacent sides in a right-angled triangle relative to a given angle.
✓By the end of this lesson students will be able to apply the Theorem of Pythagoras to find the length of an unknown side in a right-angled triangle.
✓By the end of this lesson students will be able to define and use the trigonometric ratios (sine, cosine, tangent) to find unknown side lengths in right-angled triangles.
✓By the end of this lesson students will be able to use inverse trigonometric ratios to find unknown angles in right-angled triangles.
✓By the end of this lesson students will be able to solve practical problems involving right-angled triangles using the Theorem of Pythagoras and trigonometric ratios.

Key concepts

Right-angled Triangle Terminology

A right-angled triangle is a triangle with one angle exactly 90 degrees. The sides of a right-angled triangle have special names relative to a chosen acute angle:\n- Hypotenuse: The longest side of a right-angled triangle, always opposite the 90-degree angle.\n- Opposite: The side directly across from the angle you are interested in.\n- Adjacent: The side next to the angle you are interested in, which is not the hypotenuse.

The Theorem of Pythagoras

In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is used when you know two sides of a right-angled triangle and want to find the third side.

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

Trigonometric Ratios (SOH CAH TOA)

Trigonometric ratios are ratios of the lengths of sides of a right-angled triangle, relative to an acute angle. These ratios are constant for a given angle, regardless of the size of the triangle. They are used to find unknown side lengths when an angle and one side are known.\n- Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse.\n- Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.\n- Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side.\nA useful mnemonic to remember these is SOH CAH TOA.

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

Inverse Trigonometric Ratios

Inverse trigonometric ratios are used to find the measure of an angle when you know the ratio of two sides. They are denoted by sin⁻¹, cos⁻¹, and tan⁻¹ (sometimes called arcsin, arccos, and arctan).

angle = sin⁻¹(Opposite / Hypotenuse)\nangle = cos⁻¹(Adjacent / Hypotenuse)\nangle = tan⁻¹(Opposite / Adjacent)

Key facts to remember

1A right-angled triangle has one angle of 90 degrees.
2The hypotenuse is the longest side, always opposite the 90-degree angle.
3The Theorem of Pythagoras states: a² + b² = c², where c is the hypotenuse and a and b are the other two sides.
4SOH CAH TOA is a mnemonic to remember the trigonometric ratios:\n - sin(angle) = Opposite / Hypotenuse\n - cos(angle) = Adjacent / Hypotenuse\n - tan(angle) = Opposite / Adjacent
5Use sin⁻¹, cos⁻¹, or tan⁻¹ to find an unknown angle when you know two sides.
6Always ensure your calculator is in 'DEG' (degrees) mode for trigonometry calculations.

Worked examples

Example 1

A ladder 5 metres long leans against a wall. The base of the ladder is 3 metres from the wall. How high up the wall does the ladder reach?

IDraw a diagram. The ladder, wall, and ground form a right-angled triangle.

IIIdentify the sides: Hypotenuse (c) = 5 m (ladder), one shorter side (b) = 3 m (distance from wall), unknown shorter side (a) = h (height up the wall).

IIIApply the Theorem of Pythagoras: a² + b² = c²

IVh² + 3² = 5²

Vh² + 9 = 25

VIh² = 25 - 9

VIIh² = 16

VIIIh = √16

9h = 4

Answer

The ladder reaches 4 metres up the wall.

The Theorem of Pythagoras is used when you know two sides of a right-angled triangle and need to find the third side.

Example 2

In a right-angled triangle, one angle is 30 degrees and the hypotenuse is 10 cm. Find the length of the side opposite the 30-degree angle, correct to one decimal place.

IDraw a diagram. Label the angle 30°, hypotenuse 10 cm, and the unknown side as x (opposite the 30° angle).

IIIdentify the knowns and unknowns relative to the 30° angle: Opposite = x, Hypotenuse = 10 cm.

IIIChoose the correct trigonometric ratio using SOH CAH TOA: We have Opposite and Hypotenuse, so use Sine (SOH).

IVsin(30°) = Opposite / Hypotenuse

Vsin(30°) = x / 10

VIMultiply both sides by 10: x = 10 * sin(30°)

VIICalculate using a calculator: x = 10 * 0.5

VIIIx = 5

Answer

The length of the side opposite the 30-degree angle is 5.0 cm.

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

Example 3

A ramp is 8 metres long and rises 2 metres vertically. Find the angle of elevation of the ramp to the nearest degree.

IDraw a diagram. The ramp, ground, and vertical rise form a right-angled triangle.

IIIdentify the knowns relative to the angle of elevation (let's call it θ): Opposite = 2 m (vertical rise), Hypotenuse = 8 m (ramp length).

IIIChoose the correct trigonometric ratio using SOH CAH TOA: We have Opposite and Hypotenuse, so use Sine (SOH).

IVsin(θ) = Opposite / Hypotenuse

Vsin(θ) = 2 / 8

VIsin(θ) = 0.25

VIITo find θ, use the inverse sine function: θ = sin⁻¹(0.25)

VIIICalculate using a calculator: θ ≈ 14.4775...

9Round to the nearest degree: θ ≈ 14°

Answer

The angle of elevation of the ramp is approximately 14 degrees.

Use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find unknown angles.

Common mistakes

✗Misidentifying the sides (opposite, adjacent, hypotenuse) relative to the angle being used.
✗Forgetting to set the calculator to 'DEG' (degrees) mode, leading to incorrect answers.
✗Mixing up the trigonometric ratios (e.g., using cosine instead of sine).
✗Making algebraic errors when rearranging equations to solve for an unknown.
✗Rounding intermediate steps too early, which can affect the accuracy of the final answer.

Exam tips

★Always draw a clear, labelled diagram for each problem. This helps visualise the situation and identify the knowns and unknowns.
★Clearly label the hypotenuse, opposite, and adjacent sides relative to the angle you are working with in your diagram.
★Decide whether to use the Theorem of Pythagoras (sides only) or trigonometric ratios (sides and angles) before starting calculations.
★Show all your working steps clearly. This helps you track your solution and allows for partial marks even if the final answer is incorrect.
★Before starting any trigonometry questions, always check that your calculator is in 'DEG' (degrees) mode.

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