Sin 54 Degrees

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

What is sin 54°?

Answer

sin 54° =
(√5 + 1)/4 ≈ 0.8090

Angle in radians: 54° = 3π/10. The point of the angle is in quadrant I, where sine is positive.

≈ — rounded to 4 decimal places.

P(cos 54°, sin 54°) = (√(10 − 2√5)/4, (√5 + 1)/4)

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

sin 54° = cos 36° = (√5 + 1)/4

xyOP

P(cos 54°, sin 54°) ≈ (0.5878, 0.8090)

Highlighted: the projection on the y-axis — sine, y ≈ 0.8090.

All four functions of 54°
FunctionExact valueDecimal
sin 54°(√5 + 1)/4≈ 0.8090
cos 54°√(10 − 2√5)/4≈ 0.5878
tan 54°√(25 + 10√5)/5≈ 1.3764
cot 54°√(5 − 2√5)≈ 0.7265

See Also

Recalculate with different values ↑

Sine of nearby angles

Anglesin
51°≈ 0.7771
52°≈ 0.7880
53°≈ 0.7986
54°(√5 + 1)/4
55°≈ 0.8192
56°≈ 0.8290
57°≈ 0.8387

FAQ

What is sin 54°?

sin 54° = (√5 + 1)/4 ≈ 0.8090

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 −0.5588 for sin 54?

The calculator is in radian mode (RAD): it takes sin 54 as the sine of 54 radians, sin 54 rad ≈ −0.5588. Switch it to degrees (DEG): sin 54° = (√5 + 1)/4 ≈ 0.8090. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also