Tan 105 Degrees

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

What is tan 105°?

Answer

tan 105° =
−2 − √3 ≈ −3.7321

Angle in radians: 105° = 7π/12. The point of the angle is in quadrant II, where tangent is negative.

≈ — rounded to 4 decimal places.

tan 105° = tan(180° − 75°) = −tan 75° = −2 − √3

tan 105° = sin 105° / cos 105° = ((√6 + √2)/4) / (−(√6 − √2)/4) = −2 − √3

tan 75° = tan(45° + 30°) = (1 + √3/3) / (1 − √3/3) = 2 + √3

xyOP

P(cos 105°, sin 105°) ≈ (−0.2588, 0.9659)

T(1, tan 105°) ≈ (1, −3.7321) — 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 105°
FunctionExact valueDecimal
sin 105°(√6 + √2)/4≈ 0.9659
cos 105°−(√6 − √2)/4≈ −0.2588
tan 105°−2 − √3≈ −3.7321
cot 105°√3 − 2≈ −0.2679

Reference angle: tan 75° = 2 + √3

See Also

Recalculate with different values ↑

Tangent of nearby angles

Angletan
98°≈ −7.1154
100°≈ −5.6713
101°≈ −5.1446
105°−2 − √3
106°≈ −3.4874
108°−√(5 + 2√5)
110°≈ −2.7475

FAQ

What is tan 105°?

tan 105° = −2 − √3 ≈ −3.7321

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 4.0278 for tan 105?

The calculator is in radian mode (RAD): it takes tan 105 as the tangent of 105 radians, tan 105 rad ≈ 4.0278. Switch it to degrees (DEG): tan 105° = −2 − √3 ≈ −3.7321. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also