Without a right angle, the three ratios do not apply. Two laws take over.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The Law of Sines
a ÷ sin A = b ÷ sin B = c ÷ sin C. Use it when you have a matched side-and-angle pair.
The Law of Cosines
c² = a² + b² − 2ab cos C. Use it for three sides, or two sides and the angle between them.
It contains Pythagoras
If C is 90°, cos C is 0, and the formula collapses to c² = a² + b².
Which law to use
The sine rule needs a matched side-angle pair: use it for ASA, AAS and SSA. The cosine rule handles SAS and SSS, where no pair is complete. Identifying which case you have decides the method immediately.
The sine rule
a/sin A = b/sin B = c/sin C. Each side over the sine of its opposite angle gives the same value. Put the unknown on top when solving for a side and use the reciprocal form when solving for an angle.
The cosine rule
c² = a² + b² − 2ab cos C. When C is 90°, cos C is zero and it reduces to Pythagoras — so Pythagoras is the right-angled special case of a more general law, not a separate result.
Check against the triangle
The largest angle must be opposite the longest side, and the three angles must total 180°. Both checks are quick and catch a mislabelled triangle before the answer is committed to.
Step 2: Try It Yourself
Tap and try it out.
- sin — opposite over hypotenuse0.80
- cos — adjacent over hypotenuse0.60
- tan — opposite over adjacent1.33
- The marked angle53.13°
The hypotenuse is not given. It comes from 3² + 4² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Nadia Choose a Law
A triangle has sides 7 and 9 with a 40° angle between them.
- Step 1
She has two sides and the included angle, so there is no matched side-angle pair.
Step 4: Your Turn
Practice makes it stick.
The Right Angle
Problem 1 of 2
In the Law of Cosines, what is cos 90°?
The Special Case
Problem 2 of 2
Sides 6 and 8 with a 90° angle between them. What is the third side?
Which Law
1 of 4
Sides 5 and 12 with a 90° angle between. Third side?
2 of 4
What is cos 90°?
3 of 4
Three sides known, no angles. Which law?
4 of 4
A side and its opposite angle known, plus another angle. Which law?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Sides 9 and 12 with a 90° angle between them. Third side?
Question 2 of 2
Why is Pythagoras a special case of the Law of Cosines?
What You Learned
- The Law of Sines needs a matched side and opposite angle.
- The Law of Cosines handles three sides, or two sides and the included angle.
- Pythagoras is the Law of Cosines with a right angle.