Right-angled triangles Revision Notes

In any right-angled triangle, one angle is exactly 90°. The other two angles are acute (less than 90°). For an acute angle θ, the sides of the triangle are named relative to θ: the side opposite θ, the side adjacent to θ, and the hypotenuse (the longest side opposite the right angle).

The sine, cosine, and tangent ratios relate these sides to the angle θ as follows:

Sine (sin θ) = Opposite side ÷ Hypotenuse, Cosine (cos θ) = Adjacent side ÷ Hypotenuse, Tangent (tan θ) = Opposite side ÷ Adjacent side. These ratios allow you to find unknown sides or angles when some measurements are known. Angles are measured in degrees, and answers should be rounded to one decimal place unless otherwise instructed.

• Identify the angle θ and label sides as opposite, adjacent, and hypotenuse.
• Use sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
• Ensure the calculator is set to degree mode when working with degrees.

To find an unknown side, rearrange the relevant ratio formula and substitute the known values. For example, if you know angle θ and the hypotenuse, and want the opposite side: opposite = hypotenuse × sin θ.

To find an unknown angle, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹). For example, if you know the opposite and adjacent sides, then θ = tan⁻¹(opposite ÷ adjacent).

Always check that the triangle is right-angled and that the angle used is acute before applying these ratios.

• Example: Given θ = 30° and hypotenuse = 10 cm, find opposite side.
• Calculation: opposite = 10 × sin 30° = 10 × 0.5 = 5 cm.
• Example: Given opposite = 4 cm and adjacent = 7 cm, find θ.
• Calculation: θ = tan⁻¹(4 ÷ 7) ≈ tan⁻¹(0.571) ≈ 29.7°.

Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: hypotenuse² = opposite² + adjacent².

This theorem is useful when two sides are known and the third side is unknown. It complements trigonometric methods, especially when angles are not given.

Use Pythagoras' theorem first to find missing sides, then apply trigonometric ratios to find angles, or vice versa.

• Example: Given opposite = 6 cm and adjacent = 8 cm, find hypotenuse.
• Calculation: hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
• Example: Given hypotenuse = 13 cm and adjacent = 5 cm, find opposite.
• Calculation: opposite = √(13² - 5²) = √(169 - 25) = √144 = 12 cm.

Bearings describe direction using degrees clockwise from north (0° or 360°). Problems involving bearings often require resolving distances and directions into right-angled triangles to apply trigonometry.

To solve such problems: draw a clear diagram, label bearings and distances, identify right-angled triangles, and apply Pythagoras' theorem and trigonometric ratios accordingly.

Bearings help determine angles between paths or directions, which can then be used to find unknown distances or angles.

• Example: A ship sails 30 km on a bearing of 045°, then 40 km on a bearing of 120°. Find the distance between start and end points.
• Method: Draw the path, identify the angle between legs, use cosine rule or break into right-angled triangles and apply trigonometry.
• Use bearings to find angles inside triangles, then apply sine, cosine, or tangent ratios as appropriate.

When calculating answers, round your final numerical answers to one decimal place unless instructed otherwise.

Keep extra decimal places during intermediate steps to avoid rounding errors, then round only the final answer.

Always include units in your answers (e.g., cm, m, km, degrees). Verify that your answer is reasonable given the context of the problem.

• Use a calculator set to degree mode for trigonometric calculations.
• Round final answers to one decimal place.
• Include units and label answers clearly.
• Check that calculated angles are acute and sides are positive and consistent with the triangle.