Tan 216 Degrees

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

What is tan 216°?

Answer

tan 216° =
√(5 − 2√5) ≈ 0.7265

Angle in radians: 216° = 6π/5. The point of the angle is in quadrant III, where tangent is positive.

≈ — rounded to 4 decimal places.

tan 216° = tan(180° + 36°) = tan 36° = √(5 − 2√5)

tan 216° = sin 216° / cos 216° = (−√(10 − 2√5)/4) / (−(√5 + 1)/4) = √(5 − 2√5)

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

tan 36° = sin 36° / cos 36° = √(5 − 2√5)

xyOP

P(cos 216°, sin 216°) ≈ (−0.8090, −0.5878)

T(1, tan 216°) ≈ (1, 0.7265) — where line OP meets the line x = 1, on the extension of the ray beyond O (dashed).

All four functions of 216°
FunctionExact valueDecimal
sin 216°−√(10 − 2√5)/4≈ −0.5878
cos 216°−(√5 + 1)/4≈ −0.8090
tan 216°√(5 − 2√5)≈ 0.7265
cot 216°√(25 + 10√5)/5≈ 1.3764

Reference angle: tan 36° = √(5 − 2√5)

See Also

Recalculate with different values ↑

Tangent of nearby angles

Angletan
210°√3/3
211°≈ 0.6009
215°≈ 0.7002
216°√(5 − 2√5)
225°1
227°≈ 1.0724
234°√(25 + 10√5)/5

FAQ

What is tan 216°?

tan 216° = √(5 − 2√5) ≈ 0.7265

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 −0.9695 for tan 216?

The calculator is in radian mode (RAD): it takes tan 216 as the tangent of 216 radians, tan 216 rad ≈ −0.9695. Switch it to degrees (DEG): tan 216° = √(5 − 2√5) ≈ 0.7265. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also