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Sine, Cosine, and Tangent Ratios

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. Sine, Cosine, and Tangent Ratios: A 30-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(30).
  3. Round at the end.

Answer: 30

Practice problems

1. Sine, Cosine, and Tangent Ratios: For angle theta, opposite = 24 and hypotenuse = 51. Find sin(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Sine, Cosine, and Tangent Ratios: For angle theta, opposite = 24 and hypotenuse = 51. 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 24/51.
  5. Simplify to 8/17.
  6. Check the result by substituting or estimating: the response should match 8/17 and make sense in the original problem.

Answer: 8/17

2. Sine, Cosine, and Tangent Ratios: For angle theta, adjacent = 160 and hypotenuse = 164. Find cos(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Sine, Cosine, and Tangent Ratios: For angle theta, adjacent = 160 and hypotenuse = 164. 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 160/164.
  5. Simplify to 40/41.
  6. Check the result by substituting or estimating: the response should match 40/41 and make sense in the original problem.

Answer: 40/41

3. Sine, Cosine, and Tangent Ratios: For angle theta, opposite = 100 and adjacent = 105. Find tan(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Sine, Cosine, and Tangent Ratios: For angle theta, opposite = 100 and adjacent = 105. 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 100/105.
  5. Simplify to 20/21.
  6. Check the result by substituting or estimating: the response should match 20/21 and make sense in the original problem.

Answer: 20/21

4. Sine, Cosine, and Tangent Ratios: A 30-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: Sine, Cosine, and Tangent Ratios: A 30-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(30).
  5. Round at the end.
  6. Check the result by substituting or estimating: the response should match 30 and make sense in the original problem.

Answer: 30

5. Sine, Cosine, and Tangent Ratios: A 45-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: Sine, Cosine, and Tangent Ratios: A 45-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 17 and make sense in the original problem.

Answer: 17

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