Cos 108 Degrees

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

What is cos 108°?

Answer

cos 108° =
−(√5 − 1)/4 ≈ −0.3090

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

≈ — rounded to 4 decimal places.

cos 108° = cos(180° − 72°) = −cos 72° = −(√5 − 1)/4

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

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

cos 72° = 2 cos² 36° − 1 = (√5 − 1)/4

xyOP

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

Highlighted: the projection on the x-axis — cosine, x ≈ −0.3090.

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: cos 72° = (√5 − 1)/4

See Also

Recalculate with different values ↑

Cosine of nearby angles

Anglecos
101°≈ −0.1908
105°−(√6 − √2)/4
106°≈ −0.2756
108°−(√5 − 1)/4
110°≈ −0.3420
120°−1/2
125°≈ −0.5736

FAQ

What is cos 108°?

cos 108° = −(√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 0.3755 for cos 108?

The calculator is in radian mode (RAD): it takes cos 108 as the cosine of 108 radians, cos 108 rad ≈ 0.3755. Switch it to degrees (DEG): cos 108° = −(√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