Sin 18 Degrees

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

What is sin 18°?

Answer

sin 18° =
(√5 − 1)/4 ≈ 0.3090

Angle in radians: 18° = π/10. The point of the angle is in quadrant I, where sine is positive.

≈ — rounded to 4 decimal places.

P(cos 18°, sin 18°) = (√(10 + 2√5)/4, (√5 − 1)/4)

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

sin 18° = cos 72° = 2 cos² 36° − 1 = (√5 − 1)/4

xyOP

P(cos 18°, sin 18°) ≈ (0.9511, 0.3090)

Highlighted: the projection on the y-axis — sine, y ≈ 0.3090.

All four functions of 18°
FunctionExact valueDecimal
sin 18°(√5 − 1)/4≈ 0.3090
cos 18°√(10 + 2√5)/4≈ 0.9511
tan 18°√(25 − 10√5)/5≈ 0.3249
cot 18°√(5 + 2√5)≈ 3.0777

See Also

Recalculate with different values ↑

Sine of nearby angles

Anglesin
15°(√6 − √2)/4
16°≈ 0.2756
17°≈ 0.2924
18°(√5 − 1)/4
19°≈ 0.3256
20°≈ 0.3420
21°≈ 0.3584

FAQ

What is sin 18°?

sin 18° = (√5 − 1)/4 ≈ 0.3090

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 −0.7510 for sin 18?

The calculator is in radian mode (RAD): it takes sin 18 as the sine of 18 radians, sin 18 rad ≈ −0.7510. Switch it to degrees (DEG): sin 18° = (√5 − 1)/4 ≈ 0.3090. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also