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Finding Missing Angles

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

Inverse sine, inverse cosine, and inverse tangent undo trig ratios so an angle can be found from side lengths. In this lesson, the goal is to use inverse trig to find an acute angle from side ratios. Prerequisite check: Algebra II or College Algebra foundations. Example 1: if opposite = 3 and hypotenuse = 5, then sin(theta) = 3/5. Example 2: if tan(theta) = 3/4 and the adjacent side is 20, the opposite side is 15. A common mistake is using opposite and adjacent from the wrong angle; the safer habit is to draw the right triangle, mark the angle, and name opposite, adjacent, and hypotenuse.

What you'll learn

Why it matters: Right-triangle trig supports height, distance, ramp, roof, surveying, navigation, and sight-line problems.

Worked example

Problem. Example 1 Foundation: A student uses the side opposite the other acute angle. What is the likely error?

  1. Opposite and adjacent depend on the chosen angle.
  2. Switching the angle can switch those labels.
  3. Relabel before choosing the ratio.

Answer: opposite and adjacent were labeled from the wrong angle

Practice problems

1. Practice 1 Foundation: In a right triangle for angle theta, opposite = 5 and hypotenuse = 13. What is sin(theta)?

Choices: 5/13 · 12/13 · 5/12 · 13/5

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice 1 Foundation: In a right triangle for angle theta, opposite = 5 and hypotenuse = 13. What is 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 5/13.
  5. Do not use the adjacent side for sine.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 5/13

2. Practice 2 Setup: For angle theta, adjacent = 15 and hypotenuse = 17. What is cos(theta)?

Choices: 15/17 · 8/17 · 8/15 · 17/15

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice 2 Setup: For angle theta, adjacent = 15 and hypotenuse = 17. What is 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 15/17.
  5. The side labels depend on the chosen angle.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 15/17

3. Practice 3 Meaning: Which ratio should you use to connect an angle, an opposite side, and the hypotenuse?

Choices: sine · cosine · tangent · cotangent

Show solution
  1. SOH means sine equals opposite over hypotenuse.
  2. Cosine uses adjacent and hypotenuse.
  3. Tangent uses opposite and adjacent.

Answer: sine

4. Practice 4 Method: If tan(theta) = 3/4 and the adjacent side is 8, what is the opposite side?

Choices: 6 · 8 · 10 · 5

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 4 Method: If tan(theta) = 3/4 and the adjacent side is 8, what is the opposite side?
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Tangent is opposite over adjacent.
  4. Scaling 4 to 8 multiplies by 2.
  5. The opposite side is 6.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 6

5. Practice 5 Reasoning: A right triangle has legs 15 and 36. What is the hypotenuse?

Choices: 39 · 17 · 13 · 51

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice 5 Reasoning: A right triangle has legs 15 and 36. What is the hypotenuse?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Recognize the scaled Pythagorean triple.
  4. 5-12-13 scaled by 3 gives 15-36-39.
  5. The hypotenuse is the longest side.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 39

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