Tan 165 Degrees

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

What is tan 165°?

Answer

tan 165° =
√3 − 2 ≈ −0.2679

Angle in radians: 165° = 11π/12. The point of the angle is in quadrant II, where tangent is negative.

≈ — rounded to 4 decimal places.

tan 165° = tan(180° − 15°) = −tan 15° = √3 − 2

tan 165° = sin 165° / cos 165° = ((√6 − √2)/4) / (−(√6 + √2)/4) = √3 − 2

tan 15° = tan(45° − 30°) = (1 − √3/3) / (1 + √3/3) = 2 − √3

xyOP

P(cos 165°, sin 165°) ≈ (−0.9659, 0.2588)

T(1, tan 165°) ≈ (1, −0.2679) — where line OP meets the line x = 1, on the extension of the ray beyond O (dashed).

All four functions of 165°
FunctionExact valueDecimal
sin 165°(√6 − √2)/4≈ 0.2588
cos 165°−(√6 + √2)/4≈ −0.9659
tan 165°√3 − 2≈ −0.2679
cot 165°−2 − √3≈ −3.7321

Reference angle: tan 15° = 2 − √3

See Also

Recalculate with different values ↑

Tangent of nearby angles

Angletan
160°≈ −0.3640
162°−√(25 − 10√5)/5
164°≈ −0.2867
165°√3 − 2
167°≈ −0.2309
175°≈ −0.0875
179°≈ −0.0175

FAQ

What is tan 165°?

tan 165° = √3 − 2 ≈ −0.2679

How are sin 15° and cos 75° related?

They are equal: sin α = cos(90° − α), and 15° + 75° = 90°. So sin 15° = cos 75° = (√6 − √2)/4, cos 15° = sin 75° = (√6 + √2)/4, and tan 15° = cot 75° = 2 − √3.

Why does a calculator show −15.0414 for tan 165?

The calculator is in radian mode (RAD): it takes tan 165 as the tangent of 165 radians, tan 165 rad ≈ −15.0414. Switch it to degrees (DEG): tan 165° = √3 − 2 ≈ −0.2679. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also