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Strand 2 — Geometry & Trigonometry

Right-angled Trigonometry

1st Year · 2nd Year · 3rd Year (Junior Cert)

  • ✓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 define the trigonometric ratios sine, cosine, and tangent for an acute angle in a right-angled triangle.
  • ✓By the end of this lesson students will be able to use a calculator to find the sine, cosine, and tangent of a given angle and to find an angle given its sine, cosine, or tangent.
  • ✓By the end of this lesson students will be able to apply trigonometric ratios to calculate the length of an unknown side in a right-angled triangle.
  • ✓By the end of this lesson students will be able to apply trigonometric ratios to calculate the measure of an unknown acute angle in a right-angled triangle.

Key concepts

Introduction to Trigonometry

Trigonometry is a branch of maths that studies the relationships between the sides and angles of triangles. In Junior Cycle, we focus specifically on right-angled triangles. It provides tools to find unknown side lengths or angle measures when certain information about the triangle is already known.

Labelling Sides of a Right-angled Triangle

Before applying trigonometric ratios, it is crucial to correctly label the sides of the right-angled triangle relative to the 'reference angle' (the acute angle we are working with, not the right angle). * Hypotenuse: The longest side, always opposite the right angle. * Opposite: The side directly across from the reference angle. * Adjacent: The side next to the reference angle that is not the hypotenuse.

The Sine Ratio (sin)

The sine of an acute angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

sin(angle) = Opposite / Hypotenuse
The Cosine Ratio (cos)

The cosine of an acute angle in a right-angled triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

cos(angle) = Adjacent / Hypotenuse
The Tangent Ratio (tan)

The tangent of an acute angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

tan(angle) = Opposite / Adjacent
SOH CAH TOA Mnemonic

This is a useful memory aid to remember the three primary trigonometric ratios: * SOH: Sine = Opposite / Hypotenuse * CAH: Cosine = Adjacent / Hypotenuse * TOA: Tangent = Opposite / Adjacent

Solving for Unknown Sides

To find an unknown side length in a right-angled triangle, first identify the known angle, the known side, and the unknown side. Then, choose the correct trigonometric ratio (sin, cos, or tan) that involves these three parts. Set up the equation and solve for the unknown side.

Solving for Unknown Angles

To find an unknown acute angle in a right-angled triangle, identify the two known side lengths. Choose the correct trigonometric ratio (sin, cos, or tan) that involves these two sides. Set up the equation. To find the angle, you will use the inverse trigonometric functions: sin⁻¹, cos⁻¹, or tan⁻¹. These are typically accessed by pressing 'Shift' or '2nd F' then 'sin', 'cos', or 'tan' on your calculator.

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

Key facts to remember

  • 1Trigonometry for Junior Cycle focuses exclusively on right-angled triangles.
  • 2The hypotenuse is always the longest side and is opposite the right angle.
  • 3The terms 'opposite' and 'adjacent' are relative to the chosen reference angle.
  • 4SOH CAH TOA is a crucial mnemonic for remembering the three trigonometric ratios.
  • 5sin(angle) = Opposite / Hypotenuse
  • 6cos(angle) = Adjacent / Hypotenuse
  • 7tan(angle) = Opposite / Adjacent
  • 8Use sin⁻¹, cos⁻¹, or tan⁻¹ to find the measure of an unknown angle.
  • 9Always ensure your calculator is in 'DEG' (degrees) mode for these calculations.

Worked examples

Example 1

In a right-angled triangle, the hypotenuse is 12 cm and one acute angle is 35°. Find the length of the side opposite the 35° angle, correct to one decimal place.

IDraw the triangle and label the knowns: Hypotenuse = 12 cm, Reference Angle = 35°, Unknown = Opposite side (let's call it 'x').
IIIdentify the relationship: We have the Opposite side and the Hypotenuse, and the angle. This means we use the Sine ratio (SOH).
IIIWrite the formula: sin(angle) = Opposite / Hypotenuse
IVSubstitute the values: sin(35°) = x / 12
VSolve for x: x = 12 * sin(35°)
VICalculate: x = 12 * 0.57357...
VIIRound to one decimal place.

Answer

x = 6.9 cm (correct to one decimal place)

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

Example 2

A right-angled triangle has an adjacent side of 7 cm and a hypotenuse of 10 cm relative to an acute angle θ. Find the measure of angle θ, correct to the nearest degree.

IDraw the triangle and label the knowns: Adjacent = 7 cm, Hypotenuse = 10 cm, Unknown = Angle θ.
IIIdentify the relationship: We have the Adjacent side and the Hypotenuse. This means we use the Cosine ratio (CAH).
IIIWrite the formula: cos(θ) = Adjacent / Hypotenuse
IVSubstitute the values: cos(θ) = 7 / 10
VCalculate the ratio: cos(θ) = 0.7
VISolve for θ using the inverse cosine function: θ = cos⁻¹(0.7)
VIICalculate: θ ≈ 45.5729...
VIIIRound to the nearest degree.

Answer

θ = 46° (correct to the nearest degree)

Remember to use the 'Shift' or '2nd F' button before 'cos' to get cos⁻¹.

Example 3

A vertical pole casts a shadow 5 metres long on the ground. If the angle of elevation of the sun is 40°, calculate the height of the pole, correct to two decimal places.

IDraw a right-angled triangle. The pole is the 'Opposite' side (height, let's call it 'h'). The shadow is the 'Adjacent' side (5 m). The angle of elevation is the reference angle (40°).
IIIdentify the relationship: We have the Opposite side and the Adjacent side, and the angle. This means we use the Tangent ratio (TOA).
IIIWrite the formula: tan(angle) = Opposite / Adjacent
IVSubstitute the values: tan(40°) = h / 5
VSolve for h: h = 5 * tan(40°)
VICalculate: h = 5 * 0.83909...
VIIRound to two decimal places.

Answer

h = 4.20 m (correct to two decimal places)

Always draw a diagram to help visualise the problem and label the sides correctly.

Common mistakes

  • ✗Incorrectly identifying the opposite, adjacent, or hypotenuse sides relative to the given angle.
  • ✗Using the wrong trigonometric ratio for the given information (e.g., using sin when cos is appropriate).
  • ✗Forgetting to use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹) when calculating an angle.
  • ✗Calculator not being in 'DEG' (degrees) mode, leading to incorrect numerical answers.
  • ✗Rounding intermediate calculations too early, which can affect the accuracy of the final answer.

Exam tips

  • ★Always draw a clear diagram for each problem and label all known values and the unknown you need to find.
  • ★Write down the SOH CAH TOA mnemonic at the top of your rough work to help you choose the correct ratio.
  • ★Show all steps in your working, including the formula used, the substitution of values, and the final calculation.
  • ★Before starting any trigonometry questions, double-check that your calculator is in 'DEG' (degrees) mode.
  • ★Read the question carefully to determine the required level of accuracy for the answer (e.g., one decimal place, nearest degree).

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