Sin 216 Degrees

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

What is sin 216°?

Answer

sin 216° =
−√(10 − 2√5)/4 ≈ −0.5878

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

≈ — rounded to 4 decimal places.

sin 216° = sin(180° + 36°) = −sin 36° = −√(10 − 2√5)/4

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

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

sin 36° = √(1 − cos² 36°) = √(10 − 2√5)/4

xyOP

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

Highlighted: the projection on the y-axis — sine, y ≈ −0.5878.

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: sin 36° = √(10 − 2√5)/4

See Also

Recalculate with different values ↑

Sine of nearby angles

Anglesin
210°−1/2
211°≈ −0.5150
215°≈ −0.5736
216°−√(10 − 2√5)/4
225°−√2/2
227°≈ −0.7314
234°−(√5 + 1)/4

FAQ

What is sin 216°?

sin 216° = −√(10 − 2√5)/4 ≈ −0.5878

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 0.6961 for sin 216?

The calculator is in radian mode (RAD): it takes sin 216 as the sine of 216 radians, sin 216 rad ≈ 0.6961. Switch it to degrees (DEG): sin 216° = −√(10 − 2√5)/4 ≈ −0.5878. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also