Tan 108 Degrees

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

What is tan 108°?

Answer

tan 108° =
−√(5 + 2√5) ≈ −3.0777

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

≈ — rounded to 4 decimal places.

tan 108° = tan(180° − 72°) = −tan 72° = −√(5 + 2√5)

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

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

tan 72° = cot 18° = cos 18° / sin 18° = √(5 + 2√5)

xyOP

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

T(1, tan 108°) ≈ (1, −3.0777) — where line OP meets the line x = 1, on the extension of the ray beyond O (dashed); T is beyond the drawing, the arrow points to it.

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: tan 72° = √(5 + 2√5)

See Also

Recalculate with different values ↑

Tangent of nearby angles

Angletan
101°≈ −5.1446
105°−2 − √3
106°≈ −3.4874
108°−√(5 + 2√5)
110°≈ −2.7475
120°−√3
125°≈ −1.4281

FAQ

What is tan 108°?

tan 108° = −√(5 + 2√5) ≈ −3.0777

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 2.4682 for tan 108?

The calculator is in radian mode (RAD): it takes tan 108 as the tangent of 108 radians, tan 108 rad ≈ 2.4682. Switch it to degrees (DEG): tan 108° = −√(5 + 2√5) ≈ −3.0777. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also