Trigonometry

IGCSE Edexcel Mathematics
4.1 Trigonometric Ratios
4.2 Sine & Cosine Rules
4.3 Application
Key Concepts: Trigonometry involves the relationships between angles and sides in triangles. The main ratios are sine, cosine, and tangent (SOH CAH TOA). The sine and cosine rules allow you to find missing sides and angles in non-right triangles.
Hypotenuse Adjacent Opposite θ sin θ = Opposite/Hypotenuse cos θ = Adjacent/Hypotenuse tan θ = Opposite/Adjacent

Section A: Basic Trigonometric Ratios

1. For a right-angled triangle with angle θ, the opposite side is 5 cm and the hypotenuse is 13 cm. [2]
(a) Calculate sin θ
(b) Calculate cos θ
2. A ladder leans against a wall at an angle of 65° to the ground. The ladder is 4 m long. [2]
(a) How high up the wall does the ladder reach?
(b) How far from the wall is the base of the ladder?
3. In a right-angled triangle, tan θ = 3/4. [2]
(a) Find sin θ (assuming θ is acute)
(b) Find cos θ

Section B: Sine and Cosine Rules

4. In triangle ABC: AB = 8 cm, BC = 10 cm, and angle ABC = 60°. [3]
Use the cosine rule to find AC.
5. In triangle PQR: PQ = 7 cm, QR = 9 cm, PR = 11 cm. [3]
Use the cosine rule to find angle PQR to the nearest degree.
6. In triangle XYZ: angle X = 45°, angle Y = 60°, XY = 10 cm. [3]
Use the sine rule to find YZ.

Section C: Applications and Problem Solving

7. From a point on level ground, the angle of elevation to the top of a building is 35°. After walking 50 m closer to the building, the angle of elevation becomes 50°. [4]
Calculate the height of the building.
8. Two ships leave a port at the same time. Ship A travels on a bearing of 040° at 20 km/h. Ship B travels on a bearing of 100° at 15 km/h. Calculate the distance between the ships after 2 hours. [4]
Total marks: 24

Mark Scheme

1. (a) sin θ = 5/13 = 0.385 [1]
(b) cos θ = 12/13 = 0.923 [1]
2. (a) Height = 4 × sin(65°) = 3.63 m [1]
(b) Distance = 4 × cos(65°) = 1.69 m [1]
3. If tan θ = 3/4, then opposite = 3, adjacent = 4, hypotenuse = 5 [1]
(a) sin θ = 3/5 = 0.6
(b) cos θ = 4/5 = 0.8 [1]
4. AC² = 8² + 10² - 2(8)(10)cos(60°) = 64 + 100 - 80 = 84 [2]
AC = 9.17 cm [1]
5. cos(Q) = (49 + 81 - 121)/(2 × 7 × 9) = 9/126 [2]
angle Q = 85.7° ≈ 86° [1]
6. Angle Z = 180° - 45° - 60° = 75° [1]
YZ/sin(45°) = 10/sin(75°) [1]
YZ = 10 × sin(45°)/sin(75°) = 7.32 cm [1]
7. Using two equations with tan: h/d = tan(35°) and h/(d-50) = tan(50°) [2]
Solving: d = 92.9 m, h = 65.1 m [2]
8. Distance travelled by A = 40 km, by B = 30 km [1]
Angle between directions = 100° - 40° = 60° [1]
Using cosine rule: d² = 40² + 30² - 2(40)(30)cos(60°) = 1600 + 900 - 1200 = 1300 [1]
d = 36.06 km [1]