Right Triangle Trigonometry Basics
A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.
Right triangle trigonometry compares side lengths from the viewpoint of an acute angle. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. In Triangles, Similarity, and Trigonometry, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.
What you'll learn
- Identify opposite, adjacent, and hypotenuse
- Use sine, cosine, and tangent ratios
- Solve simple right-triangle ratio problems
Worked example
Problem. In a right triangle, relative to angle A, the opposite side is 6, adjacent side is 8, and hypotenuse is 10. Find sin A.
- Sine is opposite over hypotenuse.
- sin A = 6/10.
- Simplify to 3/5.
- Connect the result back to Right Triangle Trigonometry Basics so the geometric relationship is explicit.
Answer: 3/5
Practice problems
1. A right triangle has opposite 3 and hypotenuse 5. What is sine?
Show solution
- Warm-up: First identify exactly what the question is asking: A right triangle has opposite 3 and hypotenuse 5. What is sine?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Sine = opposite/hypotenuse.
- Check the result by substituting or estimating: the response should match 3/5 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 3/5
2. A right triangle has adjacent 4 and hypotenuse 5. What is cosine?
Show solution
- Warm-up: First identify exactly what the question is asking: A right triangle has adjacent 4 and hypotenuse 5. What is cosine?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Cosine = adjacent/hypotenuse.
- Check the result by substituting or estimating: the response should match 4/5 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 4/5
3. A right triangle has opposite 6 and adjacent 8. What is tangent?
Show solution
- Core Practice: First identify exactly what the question is asking: A right triangle has opposite 6 and adjacent 8. What is tangent?
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Tangent = opposite/adjacent.
- 6/8 simplifies to 3/4.
- Check the result by substituting or estimating: the response should match 3/4 and make sense in the original problem.
Answer: 3/4
4. If sin A = 5/13 and the hypotenuse is 13, what is the opposite side?
Show solution
- Core Practice: First identify exactly what the question is asking: If sin A = 5/13 and the hypotenuse is 13, what is the opposite side?
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Sine is opposite over hypotenuse.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 5
5. If tan A = 7/24 and the adjacent side is 24, what is the opposite side?
Show solution
- Challenge: First identify exactly what the question is asking: If tan A = 7/24 and the adjacent side is 24, what is the opposite side?
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Tangent is opposite over adjacent.
- Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.
- Identify the diagram relationship, formula, theorem, or proof reason before calculating.
Answer: 7
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