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Algebra 101 - Unit 16

Lesson 106 — Advanced Trigonometric Applications

Advanced trigonometric applications involve solving real-world problems using triangles, angles of elevation and depression, bearings, and distances. This lesson emphasizes practical problem-solving with both right and oblique triangles.

Strategy for Solving Problems

  1. Identify whether the triangle is right or oblique.
  2. Label known sides, angles, and distances.
  3. Use appropriate formulas: right triangle ratios, Law of Sines, or Law of Cosines.
  4. Set up equations to solve for unknowns.
  5. Interpret the results in the context of the problem.
  6. Check solutions for feasibility and accuracy.

Examples

Example 1 — Angle of Elevation:

A person observes the top of a tower at an angle of elevation of 40° from a point 50 m away from the base. Height of the tower: h = 50·tan(40°) ≈ 42.0 m

Example 2 — Bearings:

A ship sails 20 km north and then 15 km east. Find the distance from the starting point. Use Pythagoras: d = √(20² + 15²) = √(400 + 225) = √625 = 25 km

Example 3 — Oblique Triangle (Law of Sines):

Triangle ABC, A = 50°, B = 60°, side a = 100 m. Find side b. sin A / a = sin B / b → b = a·sin B / sin A = 100·sin 60° / sin 50° ≈ 100·0.866 / 0.766 ≈ 113.0 m

Example 4 — Oblique Triangle (Law of Cosines):

Triangle with sides a = 8 m, b = 6 m, angle C = 120°. Find side c. c² = a² + b² − 2ab cos C → c² = 64 + 36 − 2·8·6·(−0.5) → c² = 100 + 48 = 148 → c ≈ 12.17 m

Practice Problems

  1. A ladder leans against a wall at an angle of 65° with the ground. If the ladder is 10 m long, how high does it reach?
  2. Two ships are 30 km apart. Ship A observes Ship B at a bearing of N 40° E. Ship B observes Ship A at a bearing of N 60° W. Find the distance between them using the Law of Sines.
  3. From a point on the ground, the angles of elevation to the top of a hill and a tree are 30° and 50° respectively, and the distance between the hill and tree is 100 m. Find the heights of both.
  4. An airplane flies 500 km on a bearing of 60° and then 300 km on a bearing of 150°. Find its distance from the starting point.
  5. Triangle ABC has sides a = 7 m, b = 9 m, and angle C = 110°. Find the third side using the Law of Cosines.