Cos 105 Degrees

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

What is cos 105°?

Answer

cos 105° =
−(√6 − √2)/4 ≈ −0.2588

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

≈ — rounded to 4 decimal places.

cos 105° = cos(180° − 75°) = −cos 75° = −(√6 − √2)/4

P(cos 105°, sin 105°) = (−(√6 − √2)/4, (√6 + √2)/4)

cos 75° = cos(45° + 30°) = (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4

xyOP

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

Highlighted: the projection on the x-axis — cosine, x ≈ −0.2588.

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: cos 75° = (√6 − √2)/4

See Also

Recalculate with different values ↑

Cosine of nearby angles

Anglecos
98°≈ −0.1392
100°≈ −0.1736
101°≈ −0.1908
105°−(√6 − √2)/4
106°≈ −0.2756
108°−(√5 − 1)/4
110°≈ −0.3420

FAQ

What is cos 105°?

cos 105° = −(√6 − √2)/4 ≈ −0.2588

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 −0.2410 for cos 105?

The calculator is in radian mode (RAD): it takes cos 105 as the cosine of 105 radians, cos 105 rad ≈ −0.2410. Switch it to degrees (DEG): cos 105° = −(√6 − √2)/4 ≈ −0.2588. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also