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Triangle Area with Trig

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

Applications ask students to choose the right trig model: Law of Sines, Law of Cosines, SSA analysis, triangle area, vectors, periodic modeling, or a cumulative mix of earlier tools.

What you'll learn

Why it matters: Surveyors, pilots, engineers, architects, and data modelers use trig when right triangles are not enough.

Worked example

Problem. Triangle Area with Trig: A triangle has sides 6, 8, and 10. Find the angle opposite side 10.

  1. Use Law of Cosines.
  2. 100 = 36 + 64 - 96cos(C).
  3. cos(C)=0, so C=90 degrees.

Answer: 90

Practice problems

1. Triangle Area with Trig: In triangle ABC, A = 40 degrees, B = 65 degrees, and a = 12. Find C.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Triangle Area with Trig: In triangle ABC, A = 40 degrees, B = 65 degrees, and a = 12. Find C.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Triangle angles sum to 180 degrees.
  4. C = 180 - 40 - 65.
  5. C = 75 degrees.
  6. Check the result by substituting or estimating: the response should match 75 and make sense in the original problem.

Answer: 75

2. Triangle Area with Trig: Use Law of Sines: A = 30 degrees, B = 45 degrees, a = 10. Find b to the nearest tenth.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Triangle Area with Trig: Use Law of Sines: A = 30 degrees, B = 45 degrees, a = 10. Find 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. Set b/sin(45) = 10/sin(30).
  4. b = 10sin(45)/sin(30).
  5. b is about 14.1.
  6. Check the result by substituting or estimating: the response should match 14.1 and make sense in the original problem.

Answer: 14.1

3. Triangle Area with Trig: Use Law of Cosines with a = 5, b = 8, and C = 60 degrees. Find c.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Triangle Area with Trig: Use Law of Cosines with a = 5, b = 8, and C = 60 degrees. Find c.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. c^2 = 25 + 64 - 2(5)(8)(1/2).
  4. c^2 = 49.
  5. c = 7.
  6. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

4. Triangle Area with Trig: A triangle has sides 6, 8, and 10. Find the angle opposite side 10.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Triangle Area with Trig: A triangle has sides 6, 8, and 10. Find the angle opposite side 10.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use Law of Cosines.
  4. 100 = 36 + 64 - 96cos(C).
  5. cos(C)=0, so C=90 degrees.
  6. Check the result by substituting or estimating: the response should match 90 and make sense in the original problem.

Answer: 90

5. Triangle Area with Trig: In SSA, why can two triangles be possible?

Choices: sine has the same positive value for an acute and an obtuse angle · cosine is never negative · all sides are equal · angles can sum over 180 degrees

Show solution
  1. Core Practice: First identify exactly what the question is asking: Triangle Area with Trig: In SSA, why can two triangles be possible?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. The Law of Sines may give sin(B).
  4. Both B and 180-B can share that sine value.
  5. Check the triangle sum.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sine has the same positive value for an acute and an obtuse angle

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