Value of tan 54°: Exact Form √(25 + 10√5)/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 54°?

√(25 + 10√5)/5, which is about 1.3764.

What is tan 54° in radians?

54° is 3π/10 radians, so tan(3π/10) = √(25 + 10√5)/5.

Is tan 54° equal to cot 36°?

Yes. tan θ = cot(90° − θ), so tan 54° = cot 36°.

How do you find tan 54° without a calculator?

tan 54° = sin 54° ÷ cos 54° = (√5 + 1)/4 ÷ √(10 − 2√5)/4. Simplify the surds: √(25 + 10√5)/5 ≈ 1.3764.