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Finding Missing Angles with Inverse Trig

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. Finding Missing Angles with Inverse Trig: A 60-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(60).
  3. Round at the end.

Answer: 52

Practice problems

1. Finding Missing Angles with Inverse Trig: For angle theta, opposite = 60 and hypotenuse = 87. Find sin(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Finding Missing Angles with Inverse Trig: For angle theta, opposite = 60 and hypotenuse = 87. Find sin(theta).
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. Sine is opposite over hypotenuse.
  4. Use 60/87.
  5. Simplify to 20/29.
  6. Check the result by substituting or estimating: the response should match 20/29 and make sense in the original problem.

Answer: 20/29

2. Finding Missing Angles with Inverse Trig: For angle theta, adjacent = 140 and hypotenuse = 148. Find cos(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Finding Missing Angles with Inverse Trig: For angle theta, adjacent = 140 and hypotenuse = 148. Find cos(theta).
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. Cosine is adjacent over hypotenuse.
  4. Use 140/148.
  5. Simplify to 35/37.
  6. Check the result by substituting or estimating: the response should match 35/37 and make sense in the original problem.

Answer: 35/37

3. Finding Missing Angles with Inverse Trig: For angle theta, opposite = 55 and adjacent = 300. Find tan(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Finding Missing Angles with Inverse Trig: For angle theta, opposite = 55 and adjacent = 300. Find tan(theta).
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. Tangent is opposite over adjacent.
  4. Use 55/300.
  5. Simplify to 11/60.
  6. Check the result by substituting or estimating: the response should match 11/60 and make sense in the original problem.

Answer: 11/60

4. Finding Missing Angles with Inverse Trig: A 60-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: Finding Missing Angles with Inverse Trig: A 60-degree angle has hypotenuse 60. Find the opposite side to the nearest tenth.
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. Use sine because opposite and hypotenuse are involved.
  4. opposite = 60sin(60).
  5. Round at the end.
  6. Check the result by substituting or estimating: the response should match 52 and make sense in the original problem.

Answer: 52

5. Finding Missing Angles with Inverse Trig: A 30-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: Finding Missing Angles with Inverse Trig: A 30-degree angle has adjacent side 12. Find the hypotenuse to the nearest tenth.
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  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 13.9 and make sense in the original problem.

Answer: 13.9

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