Tan 252 Degrees

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

What is tan 252°?

Answer

tan 252° =
√(5 + 2√5) ≈ 3.0777

Angle in radians: 252° = 7π/5. The point of the angle is in quadrant III, where tangent is positive.

≈ — rounded to 4 decimal places.

tan 252° = tan(180° + 72°) = tan 72° = √(5 + 2√5)

tan 252° = sin 252° / cos 252° = (−√(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 252°, sin 252°) ≈ (−0.3090, −0.9511)

T(1, tan 252°) ≈ (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 252°
FunctionExact valueDecimal
sin 252°−√(10 + 2√5)/4≈ −0.9511
cos 252°−(√5 − 1)/4≈ −0.3090
tan 252°√(5 + 2√5)≈ 3.0777
cot 252°√(25 − 10√5)/5≈ 0.3249

Reference angle: tan 72° = √(5 + 2√5)

See Also

Recalculate with different values ↑

Tangent of nearby angles

Angletan
227°≈ 1.0724
234°√(25 + 10√5)/5
240°√3
252°√(5 + 2√5)
255°2 + √3
270°undefined
285°−2 − √3

FAQ

What is tan 252°?

tan 252° = √(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 0.7965 for tan 252?

The calculator is in radian mode (RAD): it takes tan 252 as the tangent of 252 radians, tan 252 rad ≈ 0.7965. Switch it to degrees (DEG): tan 252° = √(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