Half a triangle: 30° and 60°
Cut an equilateral triangle in half and read the exact ratios of 30° and 60°.
An equilateral triangle cut in half
Take an equilateral triangle ABC with side 2a. Draw the perpendicular AD from A to BC.
Each angle of the triangle is 60°. The perpendicular AD cuts BC into two equal parts, so BD = a. In the right triangle ABD the angle at B is 60°, and the angle at A is 30°, half of 60°.
AB = 2a and BD = a. What is AD, from Pythagoras?
What this lesson covers
The idea
Half of an equilateral triangle of side 2a has sides a, a√3 and 2a, giving sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 and sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
An equilateral triangle cut in half
Take an equilateral triangle ABC with side 2a. Draw the perpendicular AD from A to BC.
Each angle of the triangle is 60°. The perpendicular AD cuts BC into two equal parts, so BD = a. In the right triangle ABD the angle at B is 60°, and the angle at A is 30°, half of 60°.
AB = 2a and BD = a. What is AD, from Pythagoras?
- 3a
- a√3
- a√5
Cut and read
Drop the perpendicular. Pick the length of AD. Then tap each ratio of 30° and 60° to see how it is worked out. Change a and watch what stays the same.
The 30°–60° triangle
A right triangle with angles 30° and 60° has sides a, a√3 and 2a. The shortest side a is opposite 30°.
sin 30° = a / 2a = 1/2
cos 30° = a√3 / 2a = √3 / 2
tan 30° = a / a√3 = 1/√3
For 60°, the opposite and adjacent sides swap: sin 60° = √3 / 2, cos 60° = 1/2 and tan 60° = √3. The flips are cosec 30° = 2, sec 30° = 2/√3, cot 30° = √3, cosec 60° = 2/√3, sec 60° = 2 and cot 60° = 1/√3.
Notes
A right triangle with angles 30° and 60° has sides a, a√3, 2a. So sin 30° = 1/2, cos 30° = √3 / 2, tan 30° = 1/√3, and sin 60° = √3 / 2, cos 60° = 1/2, tan 60° = √3.
Check yourself
A 30°–60°–90° triangle has shortest side 5 cm. How long is the hypotenuse?
In triangle ABC, right-angled at B, AB = 5 cm and angle ACB = 30°. How long is AC (in cm)?
Answer: 10
sin 30° = AB / AC, so 1/2 = 5 / AC and AC = 10 cm.
In the same triangle, how long is BC (in cm)? Give it as a decimal, to two places. (√3 ≈ 1.732)
Answer: 8.66
tan 30° = 5 / BC = 1/√3, so BC = 5√3 ≈ 8.66 cm.
Which pair of values is equal?
- 5√3 cm. 5√3 cm is the longer leg. The hypotenuse is twice the shortest side.
- 10 cm — correct. Yes! The sides are a, a√3, 2a. With a = 5 the hypotenuse is 2a = 10 cm.
- 15 cm. The hypotenuse is 2a, which is twice the shortest side, not three times.
- sin 30° and cos 60° — correct. Yes! Both equal 1/2. The side opposite 30° is the side adjacent to 60°.
- sin 30° and sin 60°. sin 30° = 1/2 but sin 60° = √3 / 2. They are different.
- tan 30° and tan 60°. tan 30° = 1/√3 and tan 60° = √3. They are different.