Introduction
Trigonometry is widely used to solve real-world problems involving angles and distances. Applications include navigation, engineering, physics, architecture, and surveying. This lesson will focus on solving right triangles and applying the laws of sines and cosines.
Strategy for Solving Trigonometry Problems
- Identify the triangle and label known sides and angles.
- Determine whether to use right triangle ratios (sin, cos, tan) or the law of sines/cosines.
- Set up the equation(s) based on the known values.
- Solve for the unknown angle(s) or side(s).
- Check the solution in context and round as needed.
Examples
Example 1 — Right Triangle (Height of a Building):
A surveyor measures an angle of elevation to the top of a building as 30° from a point 50 m from the base. Find the height of the building. h = 50·tan(30°) ≈ 50·0.577 ≈ 28.85 m
Example 2 — Law of Sines:
In triangle ABC, angle A = 40°, angle B = 65°, and side a = 20 m. Find side b. sin A / a = sin B / b → sin 40° / 20 = sin 65° / b → b = 20·sin 65° / sin 40° ≈ 20·0.9063 / 0.6428 ≈ 28.2 m
Example 3 — Law of Cosines:
Triangle with sides a = 8, b = 6, angle C = 60°. Find side c. c² = a² + b² − 2ab cos C → c² = 64 + 36 − 96·0.5 → c² = 100 − 48 = 52 → c ≈ 7.21
Practice Problems
- A ladder leans against a wall forming a 70° angle. If the base is 4 m from the wall, find the ladder length.
- In triangle ABC, angle A = 50°, angle B = 60°, side a = 15 m. Find side b using the law of sines.
- Triangle with sides a = 10, b = 7, and angle C = 120°. Find side c using the law of cosines.
- A flagpole casts a shadow 12 m long when the angle of elevation of the sun is 35°. Find the height of the flagpole.
- Two ships are 20 km apart. Ship A observes ship B at a 40° angle. Ship B observes ship A at a 65° angle. Find the distance between the ships using the law of sines.