A calculator gives sin 45° as 0.7071. The exact value is √2 ÷ 2, and exact values are what later courses need.
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 diagonal 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.
A pattern that helps
Write √1/2, √2/2, √3/2 for 30°, 45° and 60°. The sines climb; the cosines descend.
The 45-45-90 triangle
Legs 1 and 1, hypotenuse √2. It is half a unit square cut along the diagonal, which is exactly where the √2 comes from. So sin 45° = cos 45° = √2/2 and tan 45° = 1.
The 30-60-90 triangle
Sides 1, √3 and 2. It is half an equilateral triangle of side 2, split down the middle, which is why the shortest side is exactly half the hypotenuse. That construction makes the ratios reconstructible rather than memorised.
Exact values matter later
sin 30° = 1/2 exactly, not 0.5 as an approximation. In calculus and physics, rounding early destroys a derivation. These two triangles supply every exact value you will need for the common angles.
They populate the unit circle
Every labelled point on the unit circle comes from one of these two triangles reflected into the four quadrants. Knowing the triangles means you can rebuild the whole circle from scratch if memory fails.
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. Rosa needs the diagonal exactly.
- 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 Ramp
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
cos 60° as a decimal?
3 of 8
tan 45° as a decimal?
4 of 8
A 30-60-90 triangle has short leg 5. What is the 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
In a 30-60-90 triangle, the hypotenuse is 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 are expected to know.