A calculator gives sin 45° as 0.7071. The exact value is √2 ÷ 2, and later courses need the exact form.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The 45-45-90 triangle
Cut a square along its diagonal. Both legs stay equal and the hypotenuse is leg × √2.
The 30-60-90 triangle
Cut an equilateral triangle down the middle. The short leg is half the hypotenuse and the long leg is short leg × √3.
What they give
sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2. The cosines run the same list backwards.
Rebuild rather than memorise
Sketching the square and the equilateral triangle regenerates every value in seconds.
Where they appear
These angles turn up constantly in construction, design and every later trigonometry course.
The 45-45-90 triangle
Legs equal, hypotenuse √2 times a leg. It is half a square cut along the diagonal, which is exactly where the √2 comes from — the diagonal of a unit square.
The 30-60-90 triangle
Sides in ratio 1 : √3 : 2. It is half an equilateral triangle split down the middle, which is why the shortest side is exactly half the hypotenuse.
Exact values matter
These give sin 30° = 1/2 and cos 45° = √2/2 exactly. Exact values matter wherever a rounded decimal early in a derivation would compound into a wrong final answer.
Spotting them
Squares, equilateral triangles and regular hexagons all contain these triangles. Recognising one lets you write down lengths immediately without reaching for trigonometry.
Step 2: Try It Yourself
Tap and try it out.
- sin — opposite over hypotenuse0.71
- cos — adjacent over hypotenuse0.71
- tan — opposite over adjacent1
- The marked angle45°
The hypotenuse is not given. It comes from 5² + 5² = 50, whose square root is 7.07.
Step 3: Watch an Example
One step at a time.
Watch Rosa Find a Diagonal
A square tile has sides of 6 cm.
- Step 1
The diagonal cuts the square into two 45-45-90 triangles.
Step 4: Your Turn
Practice makes it stick.
The Square
Problem 1 of 2
A square has sides of 10. What is the diagonal, to two decimal places?
The Half Triangle
Problem 2 of 2
A 30-60-90 triangle has hypotenuse 12. What is the short leg?
Exact Values
1 of 8
sin 30° as a decimal?
2 of 8
tan 45° as a decimal?
3 of 8
cos 60° as a decimal?
4 of 8
A 30-60-90 triangle has short leg 5. Hypotenuse?
5 of 8
A 45-45-90 triangle has legs 7. Hypotenuse to two decimal places?
6 of 8
sin 60° to three decimal places?
7 of 8
Match each angle with its sine.
Tap a card on the left to start.
8 of 8
A 30-60-90 triangle has short leg 4. Long leg to two decimal places?
Step 5: Quick Check
Show what you know.
Question 1 of 2
A 30-60-90 triangle has hypotenuse 20. What is the short leg?
Question 2 of 2
Which triangle comes from cutting a square in half?
What You Learned
- The 45-45-90 triangle comes from a square; its hypotenuse is leg × √2.
- The 30-60-90 triangle comes from an equilateral triangle; the short leg is half the hypotenuse.
- These two triangles supply every exact value you need.