Tan 72 Degrees

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

What is tan 72°?

Answer

tan 72° =
√(5 + 2√5) ≈ 3.0777

Angle in radians: 72° = 2π/5. The point of the angle is in quadrant I, where tangent is positive.

≈ — rounded to 4 decimal places.

tan 72° = sin 72° / cos 72° = (√(10 + 2√5)/4) / ((√5 − 1)/4) = √(5 + 2√5)

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

tan 72° = cot 18° = cos 18° / sin 18° = √(5 + 2√5)

xyOP

P(cos 72°, sin 72°) ≈ (0.3090, 0.9511)

T(1, tan 72°) ≈ (1, 3.0777) — where line OP meets the line x = 1; T is beyond the drawing, the arrow points to it.

All four functions of 72°
FunctionExact valueDecimal
sin 72°√(10 + 2√5)/4≈ 0.9511
cos 72°(√5 − 1)/4≈ 0.3090
tan 72°√(5 + 2√5)≈ 3.0777
cot 72°√(25 − 10√5)/5≈ 0.3249

See Also

Recalculate with different values ↑

Tangent of nearby angles

Angletan
69°≈ 2.6051
70°≈ 2.7475
71°≈ 2.9042
72°√(5 + 2√5)
73°≈ 3.2709
74°≈ 3.4874
75°2 + √3

FAQ

What is tan 72°?

tan 72° = √(5 + 2√5) ≈ 3.0777

How are sin 18° and cos 72° related?

They are equal: sin α = cos(90° − α), and 18° + 72° = 90°. So sin 18° = cos 72° = (√5 − 1)/4, cos 18° = sin 72° = √(10 + 2√5)/4, and tan 18° = cot 72° = √(25 − 10√5)/5.

Why does a calculator show −0.2624 for tan 72?

The calculator is in radian mode (RAD): it takes tan 72 as the tangent of 72 radians, tan 72 rad ≈ −0.2624. Switch it to degrees (DEG): tan 72° = √(5 + 2√5) ≈ 3.0777. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also