A calculator gives sin 45° as 0.7071. The exact value is √2 ÷ 2, and exact values are what later courses need. Two triangles supply every one of them.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
An equilateral triangle has sides of 8 cm, so all three of its angles are 60°. A straight line drawn from the top corner down to the middle of the base splits it into two right triangles. Using Pythagoras alone, how long is that line, in centimetres, to two decimal places?
cm
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
The 45-45-90 triangle
Cut a square of side 1 along its diagonal. Both legs stay 1, and the diagonal is √(1 + 1) = √2. So sin 45° = 1/√2 = √2/2, cos 45° the same, tan 45° = 1.
The 30-60-90 triangle
Cut an equilateral triangle of side 2 down the middle. The short leg is 1, the hypotenuse 2, the long leg √(4 − 1) = √3. The short leg is half the hypotenuse, and the long leg is short leg × √3.
What they give
sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 = √3/3. sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3. Read them straight off the triangle.
A pattern that helps
Write √1/2, √2/2, √3/2 for the sines of 30°, 45°, 60°. The cosines run the same list backwards. Nothing to memorize beyond that.
Scaling up
A 30-60-90 triangle with hypotenuse 10 has short leg 5 and long leg 5√3 ≈ 8.66. Multiply every side of the unit triangle by the same factor.
Do not memorize the sides
Redraw the square and the equilateral triangle, and the ratios come back in seconds. Check with a calculator once: √3/2 ≈ 0.866 = sin 60°.
Step 2: Try It Yourself
Tap and try it out.
Find the exact values of sin 60°, cos 60° and tan 60° from the equilateral triangle
- 1equilateral triangle, side 2, cut down the middletwo 30-60-90 triangles
Step 0 of 4
- 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: In Real Life
A carpenter’s cuts
A 45-degree mitre on a 6-inch board has a diagonal of 6√2. A 30-degree brace of length 10 rises exactly 5. Two triangles, known by heart, save a calculator on every cut.
Step 4: Watch an Example
One step at a time.
Watch Priya Work Backwards From the Long Leg
A 30-60-90 triangle has its long leg — the side facing the 60° angle — measured at 9 cm. Priya wants the short leg and the hypotenuse exactly, without a calculator until the last line.
- Step 1
The three sides of a 30-60-90 triangle sit in the ratio 1 : √3 : 2, short leg to long leg to hypotenuse. The 9 cm side is the long leg, which is the √3 place in that list, so 9 is the short leg multiplied by √3.
Watch Dev Work Backwards From a Diagonal
A square tabletop has a diagonal of 20 cm. Dev wants the side of the square, and then its area. He writes the first line and the check at the end; you supply the three in between.
- Step 1
A square cut along its diagonal gives two 45-45-90 triangles, and the diagonal is the hypotenuse of each. In that triangle the hypotenuse is a leg times √2, so 20 = L × √2, where L is the side of the square.
Step 5: Your Turn
Practice makes it stick.
The Gate Brace
Problem 1 of 2
A garden gate is a square, 4 feet on every side. A steel brace runs corner to corner across it. The metal shop charges $15 for each foot of brace and rounds the length up to the next whole foot before charging. What does the brace cost, in dollars?
dollars
The Skate Ramp
Problem 2 of 2
A skate ramp is a 30-60-90 triangle standing on its long leg: the sloping top is the hypotenuse, 16 feet long, and the upright back is the leg facing the 30° angle. Plywood for the upright back costs $7 for each foot of its height. What does the upright back cost, in dollars?
dollars
Read It Off the Triangle
Both ratios, in both directions. Name which triangle you are in before you start.
1 of 5
A 45-45-90 triangle has legs of 9 cm. How long is the hypotenuse, in centimetres, to two decimal places?
cm
2 of 5
A 45-45-90 triangle has a hypotenuse of 14 cm. How long is each leg, in centimetres, to two decimal places?
cm
3 of 5
A 30-60-90 triangle has a short leg of 7 cm. How long is the hypotenuse, in centimetres?
cm
4 of 5
A 30-60-90 triangle has a hypotenuse of 18 cm. How long is the long leg, the side facing the 60° angle, in centimetres, to two decimal places?
cm
5 of 5
A 30-60-90 triangle has short leg 1, long leg √3 and hypotenuse 2. Reading the ratio straight off that triangle, what is cos 30° to three decimal places?
Whose Step?
One line in each piece of working is a slip. Find it, then fix it and finish the job.
1 of 6
A 30-60-90 triangle has a hypotenuse of 20 cm. Maya worked out the side facing the 60° angle and got 10 cm. One line is a slip. Which?
Maya’s working
- 1hypotenuse = 20 cm
- 2the side facing 60° is half the hypotenuse
- 3side facing 60° = 20 ÷ 2
- Answerside facing 60° = 10 cm
2 of 6
Fix Maya’s working and finish the job. How long is the side facing the 60° angle, in centimetres, to two decimal places?
cm
3 of 6
A 30-60-90 triangle has a long leg, the side facing the 60° angle, of 12 cm. Leo worked out the hypotenuse. One line is a slip. Which?
Leo’s working
- 1long leg = 12 cm
- 2short leg = 12 × √3 ≈ 20.78 cm× √3
- 3hypotenuse = 2 × 20.78
- Answerhypotenuse ≈ 41.57 cm
4 of 6
Fix Leo’s working and finish the job. How long is the hypotenuse, in centimetres, to two decimal places?
cm
5 of 6
A square has a diagonal of 16 cm. Ana worked out the side of the square. One line is a slip. Which?
Ana’s working
- 1the diagonal cuts the square into two 45-45-90 triangles
- 2diagonal = side × √2
- 3side = 16 × √2× √2
- Answerside ≈ 22.63 cm
6 of 6
Fix Ana’s working and finish the job. How long is the side of the square, in centimetres, to two decimal places?
cm
Pick the Move
Decide the first move, and why, before any number is worked out.
1 of 2
A ladder leans against a wall, making an angle of 60° with the ground, and its foot stands 4 feet from the wall. The length of the ladder is wanted, without a calculator. Which first move gets there?
2 of 2
A 45-45-90 triangle has a hypotenuse of 30 cm, and both legs are wanted. Which first move gets there?
Same Value?
No triangle needs solving here. Judge each one by the ratio it comes from.
1 of 2
Select every expression below that has the same value as sin 45°.
2 of 2
Select every triangle below whose three sides are in the ratio 1 : √3 : 2, which is what makes a triangle a 30-60-90 one.
Step 6: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A 30-60-90 triangle and a 45-45-90 triangle are drawn with the same hypotenuse, 12 cm. How much longer is a leg of the 45-45-90 triangle than the short leg of the 30-60-90 triangle, in centimetres, to two decimal places (take √2 as 1.4142)?
cm
Stretch 2 of 2
A 30-60-90 triangle is to be drawn so that its long leg is exactly 5 cm longer than its short leg. Only one size of triangle does this. How long is the short leg, in centimetres, to two decimal places (take √3 as 1.7321)?
cm
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 and the long leg is short leg × √3.
- The sines of 30°, 45°, 60° are √1/2, √2/2, √3/2; the cosines run backwards.
- Redraw the two triangles rather than memorizing a table.
For the grown-up
The misconception this lesson targets is that the 2 and the √2 belong to the same triangle: a student halves a 45-45-90 hypotenuse to get a leg, or doubles a 45-45-90 leg to get the diagonal. Doubling is the 30-60-90 rule and √2 is the 45-45-90 rule, and each belongs to the shape it was cut from — a square gives the √2, a halved equilateral triangle gives the 2 and the √3. The quieter cousin is running a ratio backwards: multiplying by √3 where dividing was needed, which always shows up as the sides landing out of order: a short leg longer than the long leg, or a leg longer than the hypotenuse. One question catches both — which side faces the smallest angle? — because the three sides must rise in the same order as the three angles they face.