All right triangles with the same acute angle are similar, so their side ratios are identical.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The three ratios have names
Sine is opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent.
The ratio depends only on the angle
That is why a table of values works at all. Scale the triangle and nothing changes.
Solving a triangle
Pick the ratio that involves the two sides you care about, then solve for the unknown.
Similarity is why the ratios work
All right triangles with the same acute angle are similar by AA, so their side ratios are identical. That is what makes sine, cosine and tangent depend on the angle alone rather than on the size of the triangle.
The three definitions
Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. Opposite and adjacent are relative to the angle you chose, so identify that angle before labelling anything.
The hypotenuse never changes role
The hypotenuse is always the side opposite the right angle, regardless of which acute angle you are working with. Only "opposite" and "adjacent" swap when you switch angles, which is a useful anchor.
What values are possible
Sine and cosine of an acute angle are always between 0 and 1, since the hypotenuse is the longest side. Tangent can be any positive number. An answer outside those ranges signals a mislabelled triangle.
Step 2: Try It Yourself
Change a leg and watch the ratios move with the angle.
- sin — opposite over hypotenuse0.60
- cos — adjacent over hypotenuse0.80
- tan — opposite over adjacent0.75
- The marked angle36.87°
The hypotenuse is not given. It comes from 4² + 3² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Sam Scale a Triangle
Sam compares a 3-4-5 triangle with a 6-8-10 triangle.
- Step 1
In the first, the ratio opposite over hypotenuse is 3/5.
Step 4: Your Turn
Practice makes it stick.
The Ratio
Problem 1 of 2
A right triangle has legs 3 and 4. What is the hypotenuse?
The Tangent
Problem 2 of 2
Opposite 6, adjacent 8. What is the tangent, as a decimal?
Ratios in Right Triangles
1 of 4
Legs 6 and 8. Hypotenuse?
2 of 4
Opposite 3, hypotenuse 5. Sine, as a decimal?
3 of 4
Adjacent 4, hypotenuse 5. Cosine, as a decimal?
4 of 4
Does scaling a triangle change its sine?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Legs 5 and 12. Hypotenuse?
Question 2 of 2
Why does the sine of an angle not depend on the triangle size?
What You Learned
- Sine, cosine, and tangent are ratios of sides.
- They depend on the angle alone, because such triangles are all similar.
- Choose the ratio containing the sides you know and want.