Tan 54 Degrees

Sin, Cos, Tan, Cot of an Angle
Function
Angle (°)

What is tan 54°?

Answer

tan 54° =
√(25 + 10√5)/5 ≈ 1.3764

Angle in radians: 54° = 3π/10. The point of the angle is in quadrant I, where tangent is positive.

≈ — rounded to 4 decimal places.

tan 54° = sin 54° / cos 54° = ((√5 + 1)/4) / (√(10 − 2√5)/4) = √(25 + 10√5)/5

Regular pentagon (golden ratio): cos 36° = (√5 + 1)/4.

tan 54° = cot 36° = cos 36° / sin 36° = √(25 + 10√5)/5

xyOP

P(cos 54°, sin 54°) ≈ (0.5878, 0.8090)

T(1, tan 54°) ≈ (1, 1.3764) — where line OP meets the line x = 1.

All four functions of 54°
FunctionExact valueDecimal
sin 54°(√5 + 1)/4≈ 0.8090
cos 54°√(10 − 2√5)/4≈ 0.5878
tan 54°√(25 + 10√5)/5≈ 1.3764
cot 54°√(5 − 2√5)≈ 0.7265

See Also

Recalculate with different values ↑

Tangent of nearby angles

Angletan
51°≈ 1.2349
52°≈ 1.2799
53°≈ 1.3270
54°√(25 + 10√5)/5
55°≈ 1.4281
56°≈ 1.4826
57°≈ 1.5399

FAQ

What is tan 54°?

tan 54° = √(25 + 10√5)/5 ≈ 1.3764

How are sin 36° and cos 54° related?

They are equal: sin α = cos(90° − α), and 36° + 54° = 90°. So sin 36° = cos 54° = √(10 − 2√5)/4, cos 36° = sin 54° = (√5 + 1)/4, and tan 36° = cot 54° = √(5 − 2√5).

Why does a calculator show 0.6738 for tan 54?

The calculator is in radian mode (RAD): it takes tan 54 as the tangent of 54 radians, tan 54 rad ≈ 0.6738. Switch it to degrees (DEG): tan 54° = √(25 + 10√5)/5 ≈ 1.3764. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also