Value of tan 72°: Exact Form √(5 + 2√5) & Decimal

How to derive it

Angle	sin	cos	tan
15°	(√6 − √2)/4	(√6 + √2)/4	2 − √3
18°	(√5 − 1)/4	√(10 + 2√5)/4	√(25 − 10√5)/5
22.5°	√(2 − √2)/2	√(2 + √2)/2	√2 − 1
30°	1/2	√3/2	1/√3
36°	√(10 − 2√5)/4	(√5 + 1)/4	√(5 − 2√5)
45°	1/√2	1/√2	1
54°	(√5 + 1)/4	√(10 − 2√5)/4	√(25 + 10√5)/5
60°	√3/2	1/2	√3
67.5°	√(2 + √2)/2	√(2 − √2)/2	√2 + 1
72°	√(10 + 2√5)/4	(√5 − 1)/4	√(5 + 2√5)
75°	(√6 + √2)/4	(√6 − √2)/4	2 + √3

For whole-degree values from 0° to 90°, see the trigonometry table.

What is the value of tan 72°?

√(5 + 2√5), which is about 3.0777.

What is tan 72° in radians?

72° is 2π/5 radians, so tan(2π/5) = √(5 + 2√5).

Is tan 72° equal to cot 18°?

Yes. tan θ = cot(90° − θ), so tan 72° = cot 18°.

How do you find tan 72° without a calculator?

tan 72° = sin 72° ÷ cos 72° = √(10 + 2√5)/4 ÷ (√5 − 1)/4. Simplify the surds: √(5 + 2√5) ≈ 3.0777.