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O-Level Mathematics · Secondary 3 and 4

Trigonometry

Trigonometry links the sides and angles of triangles. In O-Level Mathematics you use Pythagoras' theorem and the ratios sine, cosine and tangent in right-angled triangles, extend sine and cosine to obtuse angles, and use the sine rule, the cosine rule and the area formula ½ab sin C in any triangle. These tools are applied to angles of elevation and depression, bearings and simple three-dimensional problems. Trigonometry also supports mensuration and vectors, and it is developed much further in Additional Mathematics.

What the syllabus expects

  • Use Pythagoras' theorem and the sine, cosine and tangent ratios to find sides and angles in right-angled triangles
  • Find the sine and cosine of obtuse angles using sin(180° − θ) = sin θ and cos(180° − θ) = −cos θ
  • Use the sine rule and the cosine rule to find unknown sides and angles in any triangle
  • Find the area of a triangle using ½ab sin C
  • Solve problems involving angles of elevation and depression and bearings
  • Solve simple problems in three dimensions by identifying the right-angled triangles involved

Right-angled triangles

In a right-angled triangle, label the sides relative to the angle θ you are using: the hypotenuse is opposite the right angle, the opposite side faces θ and the adjacent side is next to θ. Then sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent, often remembered as SOH CAH TOA. Pythagoras' theorem, a² + b² = c² with c the hypotenuse, links the three sides without needing any angle. To find a side, choose the ratio that contains the side you want and the side you know. For a hypotenuse of 10 cm and an angle of 35°, the opposite side is 10 sin 35° = 5.74 cm. To find an angle, use the inverse function: if tan θ = 7/4, then θ = tan⁻¹(7/4) = 60.3°. Make sure your calculator is in degree mode.

Obtuse angles

Sine and cosine are also defined for angles between 90° and 180°, which you need for the sine rule, the cosine rule and the area formula. The key results are sin(180° − θ) = sin θ and cos(180° − θ) = −cos θ. So sin 150° = sin 30° = 0.5 and cos 120° = −cos 60° = −0.5. The sign of the cosine tells you whether an angle is acute or obtuse: a negative cosine means an obtuse angle. This is why the cosine rule never gives an ambiguous angle. The sine is positive for both acute and obtuse angles, so if sin θ = 0.6, θ could be 36.9° or 180° − 36.9° = 143.1°. Your calculator only gives the acute value, so when you use the sine rule to find an angle you must decide from the diagram or the other information whether the obtuse angle is the one you need.

Sine rule, cosine rule and area

For any triangle ABC with sides a, b and c opposite angles A, B and C, the sine rule is a/sin A = b/sin B = c/sin C. Use it when you know a side and the angle opposite it, plus one more side or angle. The cosine rule is a² = b² + c² − 2bc cos A. Use it when you know two sides and the angle between them to find the third side, or all three sides to find an angle, rearranged as cos A = (b² + c² − a²)/(2bc). The area of any triangle is ½ab sin C, using two sides and the angle between them. A good habit is to sketch the triangle, mark what you know and ask whether any pair of a side and its opposite angle is complete. If one pair is complete, the sine rule is usually the way in. If not, start with the cosine rule.

Elevation, depression, bearings and 3D

An angle of elevation is measured upwards from the horizontal to the line of sight, and an angle of depression is measured downwards from the horizontal. Both are measured from the horizontal, never from the vertical, and the angle of elevation of B from A equals the angle of depression of A from B because they are alternate angles. A bearing is an angle measured clockwise from north and written with three figures, such as 060° or 245°. If the bearing of Q from P is θ, the bearing of P from Q is θ + 180°, or θ − 180° if θ is more than 180°. Draw a north line at every point you work from. In three-dimensional problems, such as finding the angle between a line and a plane, draw the right-angled triangle you need as a separate flat diagram, find any missing length with Pythagoras' theorem first, and then apply trigonometry.

Worked example

In triangle ABC, AB = 8 cm, AC = 11 cm and angle BAC = 50°. Find (a) the length of BC, (b) the area of triangle ABC, (c) angle ABC.

  1. Two sides and the included angle are known, so use the cosine rule: BC² = 8² + 11² − 2(8)(11) cos 50°.
  2. BC² = 64 + 121 − 176 cos 50° = 185 − 113.131 = 71.869.
  3. BC = √71.869 = 8.4776 cm, which is 8.48 cm to 3 significant figures.
  4. Area = ½ × 8 × 11 × sin 50° = 44 × 0.76604 = 33.706 cm².
  5. For angle ABC, use the cosine rule with all three sides: cos B = (8² + 8.4776² − 11²)/(2 × 8 × 8.4776) = 14.870/135.64 = 0.10963.
  6. B = cos⁻¹(0.10963) = 83.7°. Using the cosine rule here avoids having to decide between an acute and an obtuse answer.

Answer: (a) BC = 8.48 cm (b) Area = 33.7 cm² (c) Angle ABC = 83.7°

Common mistakes

  • Using radian mode. If sin 30° comes out as −0.988, the calculator is in radians. Check the mode at the start of every paper.
  • Using the sine rule to find an angle that may be obtuse. sin⁻¹ only returns acute angles, so an angle that should be 110° may come out as 70°. When the angle could be obtuse, use the cosine rule instead, or check whether it is opposite the longest side.
  • Calculating the cosine rule in the wrong order. b² + c² − 2bc cos A must be evaluated as written. Working out (b² + c² − 2bc) first and then multiplying by cos A gives a wrong answer.
  • Measuring a bearing from the wrong direction. Bearings are measured clockwise from north and written with three figures, so a direction 40° east of north is 040°, not 40° or 320°.
  • Measuring angles of elevation from the vertical. The angle is between the horizontal and the line of sight. Draw the horizontal line on your diagram before you label the angle.
  • Rounding lengths before using them again. Rounding BC to 8.5 before finding angle B shifts the answer to 83.6°. Keep at least 4 or 5 significant figures in intermediate values.

Exam tips

  • Before choosing a rule, mark on your sketch which sides and angles you know. Two sides and the angle between them points to the cosine rule, and a complete side and opposite angle pair points to the sine rule.
  • In bearings questions, draw a north line at every point and use the parallel north lines to find angles inside the triangle, using alternate or co-interior angles.
  • Give angles to 1 decimal place and lengths to 3 significant figures unless the question says otherwise, but keep more figures in your working.

Practice questions

1. A ladder 5 m long leans against a vertical wall and makes an angle of 70° with the horizontal ground. How high up the wall does it reach?

Show answer

4.70 m. The ladder is the hypotenuse and the height is opposite the 70° angle, so height = 5 sin 70° = 5 × 0.93969 = 4.698 m.

2. Given that θ is obtuse and sin θ = 0.6, find θ and cos θ.

Show answer

θ = 143.1° and cos θ = −0.8. The acute angle with sine 0.6 is sin⁻¹(0.6) = 36.87°, so θ = 180° − 36.87° = 143.13°. Then cos θ = −cos 36.87° = −0.8.

3. A ship sails 12 km from P to Q on a bearing of 060°, then 9 km from Q to R on a bearing of 150°. Find the distance PR and the bearing of R from P.

Show answer

PR = 15 km and the bearing is 096.9°. The bearing of P from Q is 240°, so angle PQR = 240° − 150° = 90°. By Pythagoras' theorem, PR = √(12² + 9²) = √225 = 15 km. Angle QPR = tan⁻¹(9/12) = 36.87°, so the bearing of R from P is 060° + 36.87° = 096.9°.

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