Right-triangle ratios
Connect an acute angle to ratios of right-triangle sides.
Right-triangle ratios is one of 1 right triangles formula in the trigonometry section of this library, and it is used at high school level.
Why right-triangle ratios works
Any two right triangles that share an acute angle are similar, so every pair of matching sides shrinks or grows by the same factor. That factor cancels when you divide one side by another, which leaves a number that depends only on the angle. That is why a 3-4-5 triangle and a 30-40-50 triangle both give sine 0.6.
What each symbol means
opp and adj are relative to $\theta$; hyp is opposite the right angle.
Right-triangle ratios: when it holds
Use a right triangle and a consistent angle mode.
When it stops applying
These three ratios only describe an acute angle inside a right triangle, so they stop short at 90 degrees. An angle of 120 degrees has no right triangle to sit in, and you need the unit circle to get sine 0.866 and cosine −0.5 for it.
Right-triangle ratios: a worked example
With opposite $3$ and hypotenuse $5$, $\sin\theta=3/5$.
The mistake to avoid
What people do: Students label one side opposite and one adjacent from the picture, then keep those labels when the question switches to the other acute angle.
Why it goes wrong: Opposite and adjacent are not properties of a side. They describe where a side sits relative to the angle you picked, so they trade places when the angle changes.
Do this instead: Write the angle you are using inside the triangle first, then trace: the leg that touches that angle is adjacent, the leg across from it is opposite, and the long side facing the right angle is always the hypotenuse.
Right-triangle ratios: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. opp and adj are relative to $\theta$; hyp is opposite the right angle.
- Check the conditions before substituting. Use a right triangle and a consistent angle mode.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Right triangles
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where right-triangle ratios comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Unit Circle — the lesson behind this formula: connect angles to coordinates, sine, and cosine.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about right-triangle ratios
Can sine or cosine ever come out bigger than 1?
No. Both divide a leg by the hypotenuse, and the hypotenuse is the longest side of a right triangle, so the fraction is always less than 1. Tangent has no such cap because it compares two legs.
What happens to the ratios if I double every side of the triangle?
Nothing changes. Doubling turns 3/5 into 6/10, which is the same number. Size does not matter to these three ratios, only the angle does, and that is the whole reason one table of values serves every right triangle.
Both acute angles are marked in my triangle, so which one do I use?
Use the angle that appears in the part of the question you are answering. Solve the problem twice with the two angles and you get the same triangle described two ways, since the two acute angles add to 90 degrees.
My calculator gives sin 30 as −0.988 instead of 0.5, what went wrong?
It is set to radians and read your 30 as 30 radians, roughly four and three quarter turns around the circle. Switch to degree mode and the same keystrokes return 0.5.