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What Trigonometry Measures

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

Right-triangle trigonometry connects an acute angle to side ratios. Students label opposite, adjacent, and hypotenuse from the chosen angle, then use sine, cosine, tangent, or inverse trig to compute a missing side, missing angle, height, distance, or sight line.

What you'll learn

Why it matters: Surveying, ramps, roof pitch, ladder safety, navigation, and line-of-sight estimates all use right-triangle trig.

Worked example

Problem. What Trigonometry Measures: A 45-degree angle has hypotenuse 60. Find the opposite side to the nearest tenth.

  1. Use sine because opposite and hypotenuse are involved.
  2. opposite = 60sin(45).
  3. Round at the end.

Answer: 42.4

Practice problems

1. What Trigonometry Measures: For angle theta, opposite = 15 and hypotenuse = 39. Find sin(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: What Trigonometry Measures: For angle theta, opposite = 15 and hypotenuse = 39. Find sin(theta).
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Sine is opposite over hypotenuse.
  4. Use 15/39.
  5. Simplify to 5/13.
  6. Check the result by substituting or estimating: the response should match 5/13 and make sense in the original problem.

Answer: 5/13

2. What Trigonometry Measures: For angle theta, adjacent = 96 and hypotenuse = 100. Find cos(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: What Trigonometry Measures: For angle theta, adjacent = 96 and hypotenuse = 100. Find cos(theta).
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Cosine is adjacent over hypotenuse.
  4. Use 96/100.
  5. Simplify to 24/25.
  6. Check the result by substituting or estimating: the response should match 24/25 and make sense in the original problem.

Answer: 24/25

3. What Trigonometry Measures: For angle theta, opposite = 40 and adjacent = 75. Find tan(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: What Trigonometry Measures: For angle theta, opposite = 40 and adjacent = 75. Find tan(theta).
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Tangent is opposite over adjacent.
  4. Use 40/75.
  5. Simplify to 8/15.
  6. Check the result by substituting or estimating: the response should match 8/15 and make sense in the original problem.

Answer: 8/15

4. What Trigonometry Measures: A 45-degree angle has hypotenuse 60. Find the opposite side to the nearest tenth.

Show solution
  1. Core Practice: First identify exactly what the question is asking: What Trigonometry Measures: A 45-degree angle has hypotenuse 60. Find the opposite side to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use sine because opposite and hypotenuse are involved.
  4. opposite = 60sin(45).
  5. Round at the end.
  6. Check the result by substituting or estimating: the response should match 42.4 and make sense in the original problem.

Answer: 42.4

5. What Trigonometry Measures: A 60-degree angle has adjacent side 12. Find the hypotenuse to the nearest tenth.

Show solution
  1. Core Practice: First identify exactly what the question is asking: What Trigonometry Measures: A 60-degree angle has adjacent side 12. Find the hypotenuse to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use cosine because adjacent and hypotenuse are involved.
  4. cos(theta) = adjacent/hypotenuse.
  5. Divide by cos(theta).
  6. Check the result by substituting or estimating: the response should match 24 and make sense in the original problem.

Answer: 24

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