All right triangles with the same acute angle are similar, so their side ratios are identical. The angle alone determines the ratios.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Naming the sides
The hypotenuse is opposite the right angle. The other two are named relative to the angle you are using.
The names move
A side that is opposite one acute angle is adjacent to the other. Fix your angle before naming anything.
The three ratios
Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.
A quick check
Sine and cosine are always below 1, because the hypotenuse is the longest side. A sine above 1 is an error.
What they are for
They connect an angle to a length, which is what lets a measured angle produce an unreachable distance.
Similarity is what makes the ratios work
All right triangles with a given acute angle are similar by AA, so their side ratios are identical. That is why sine, cosine and tangent depend only on the angle and not on the size.
The three ratios
Sine is opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent. Opposite and adjacent depend on which acute angle you chose, so identify it first.
The hypotenuse never changes role
It is always opposite the right angle, whichever acute angle you work from. Only the other two labels swap, which makes the hypotenuse a reliable anchor when labelling.
The values are bounded
Sine and cosine of an acute angle lie strictly between 0 and 1, because the hypotenuse is the longest side. A value outside that range means the triangle was mislabelled.
Step 2: Try It Yourself
Tap and try it out.
- 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 Ines Compute the Ratios
A right triangle has opposite 3, adjacent 4 and hypotenuse 5.
- Step 1
Sine is opposite over hypotenuse: 3 ÷ 5 = 0.6.
Step 4: Your Turn
Practice makes it stick.
The Sine
Problem 1 of 2
Opposite 3, hypotenuse 5. What is the sine, as a decimal?
The Tangent
Problem 2 of 2
Opposite 3, adjacent 4. What is the tangent, as a decimal?
Name and Compute
1 of 8
Adjacent 4, hypotenuse 5. Cosine as a decimal?
2 of 8
Opposite 8, hypotenuse 10. Sine as a decimal?
3 of 8
Legs 6 and 8. What is the hypotenuse?
4 of 8
Opposite 6, adjacent 8. Tangent as a decimal?
5 of 8
Can a sine ever exceed 1? 1 yes, 0 no.
6 of 8
Legs 5 and 12. What is the hypotenuse?
7 of 8
Match each ratio with its definition.
Tap a card on the left to start.
8 of 8
Opposite 5, hypotenuse 13. Sine to two decimal places?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Adjacent 12, hypotenuse 13. Cosine to two decimal places?
Question 2 of 2
Why does one angle determine the ratios?
What You Learned
- The acute angle alone determines the side ratios, because such triangles are all similar.
- Sine is opposite over hypotenuse, cosine adjacent over hypotenuse, tangent opposite over adjacent.
- Sine and cosine are always below 1.