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Cumulative Trig Modeling

A free Trigonometry lesson from the “Applications of Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

This lesson is a cumulative review where each problem could need a different trig tool, so the first decision is always which method fits the given information. Right-triangle ratios work with a right angle; the Law of Sines needs an angle-side pair; the Law of Cosines fits SAS or SSS; the area formula (1/2)ab*sin(C) needs two sides and the included angle; vectors use components and magnitude; and periodic models use sinusoidal functions for repeating quantities. Reading the givens correctly to pick the matching tool is the core skill being practiced.

What you'll learn

Why it matters: Engineers, surveyors, and navigators rarely get told which formula to use; they read a real situation, decide whether it is a triangle, a force, or a repeating cycle, and reach for the matching trig tool.

Worked example

Problem. A surveyor measures two sides of a triangular plot as 30 m and 40 m with a 110 degree angle between them. Find the length of the third side and the area of the plot, each to the nearest tenth.

  1. Two sides and the included angle (SAS) means use the Law of Cosines for the third side: c^2 = 30^2 + 40^2 - 2(30)(40)cos(110) = 2500 - 2400(-0.342) ~ 3321, so c ~ 57.6 m
  2. SAS also fits the area formula: Area = (1/2)(30)(40)sin(110) ~ 600(0.9397) ~ 563.8 m^2
  3. Both results use the same two sides and included angle, just different formulas.

Answer: Third side ~ 57.6 m; area ~ 563.8 m^2

Practice problems

1. From a point 40 m from the base of a tower, the angle of elevation to the top is 35 degrees. How tall is the tower, to the nearest tenth of a meter?

Show solution
  1. Warm-up: First identify exactly what the question is asking: From a point 40 m from the base of a tower, the angle of elevation to the top is 35 degrees. How tall is the tower, to the nearest tenth of a meter?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Right triangle: the 40 m is adjacent and the height is opposite, so use tangent.
  4. tan(35) = height / 40, so height = 40*tan(35) ~ 28.0 m.
  5. Check the result by substituting or estimating: the response should match 28.0 m and make sense in the original problem.

Answer: 28.0 m

2. A ladder leans against a wall with its base 3 m from the wall and its top reaching along a 10 m ladder. What angle does the ladder make with the ground, to the nearest tenth of a degree?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A ladder leans against a wall with its base 3 m from the wall and its top reaching along a 10 m ladder. What angle does the ladder make with the ground, to the nearest tenth of a degree?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Right triangle: the 3 m is adjacent to the ground angle and 10 m is the hypotenuse, so use cosine.
  4. cos(theta) = 3/10, so theta = arccos(0.3) ~ 72.5 degrees.
  5. Check the result by substituting or estimating: the response should match 72.5 degrees and make sense in the original problem.

Answer: 72.5 degrees

3. A triangle gives you two sides and the angle between them and asks for the third side. Which tool fits?

Choices: Law of Sines · Law of Cosines · Right-triangle tangent ratio · The periodic model formula

Show solution
  1. Warm-up: First identify exactly what the question is asking: A triangle gives you two sides and the angle between them and asks for the third side. Which tool fits?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Two sides and the included angle is the SAS case.
  4. The Law of Cosines is built for SAS (and SSS) triangles.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Law of Cosines

4. In a triangle, angle A = 40 degrees, angle B = 75 degrees, and side a = 12 (opposite A). Find side b, to the nearest tenth.

Show solution
  1. Core Practice: First identify exactly what the question is asking: In a triangle, angle A = 40 degrees, angle B = 75 degrees, and side a = 12 (opposite A). Find side b, to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. An angle-side pair plus another angle means use the Law of Sines.
  4. b/sin(75) = 12/sin(40), so b = 12*sin(75)/sin(40) ~ 18.0.
  5. Check the result by substituting or estimating: the response should match 18.0 and make sense in the original problem.

Answer: 18.0

5. A triangle has sides a = 8 and b = 11 with included angle C = 37 degrees. Find side c, to the nearest tenth.

Show solution
  1. Core Practice: First identify exactly what the question is asking: A triangle has sides a = 8 and b = 11 with included angle C = 37 degrees. Find side c, to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Two sides and the included angle (SAS) calls for the Law of Cosines.
  4. c^2 = 8^2 + 11^2 - 2(8)(11)cos(37) ~ 44.4, so c ~ 6.7.
  5. Check the result by substituting or estimating: the response should match 6.7 and make sense in the original problem.

Answer: 6.7

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