Sin 108 Degrees

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

What is sin 108°?

Answer

sin 108° =
√(10 + 2√5)/4 ≈ 0.9511

Angle in radians: 108° = 3π/5. The point of the angle is in quadrant II, where sine is positive.

≈ — rounded to 4 decimal places.

sin 108° = sin(180° − 72°) = sin 72° = √(10 + 2√5)/4

P(cos 108°, sin 108°) = (−(√5 − 1)/4, √(10 + 2√5)/4)

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

sin 72° = cos 18° = √(1 − sin² 18°) = √(10 + 2√5)/4

xyOP

P(cos 108°, sin 108°) ≈ (−0.3090, 0.9511)

Highlighted: the projection on the y-axis — sine, y ≈ 0.9511.

All four functions of 108°
FunctionExact valueDecimal
sin 108°√(10 + 2√5)/4≈ 0.9511
cos 108°−(√5 − 1)/4≈ −0.3090
tan 108°−√(5 + 2√5)≈ −3.0777
cot 108°−√(25 − 10√5)/5≈ −0.3249

Reference angle: sin 72° = √(10 + 2√5)/4

See Also

Recalculate with different values ↑

Sine of nearby angles

Anglesin
101°≈ 0.9816
105°(√6 + √2)/4
106°≈ 0.9613
108°√(10 + 2√5)/4
110°≈ 0.9397
120°√3/2
125°≈ 0.8192

FAQ

What is sin 108°?

sin 108° = √(10 + 2√5)/4 ≈ 0.9511

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 0.9268 for sin 108?

The calculator is in radian mode (RAD): it takes sin 108 as the sine of 108 radians, sin 108 rad ≈ 0.9268. Switch it to degrees (DEG): sin 108° = √(10 + 2√5)/4 ≈ 0.9511. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also