Sin 105 Degrees

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

What is sin 105°?

Answer

sin 105° =
(√6 + √2)/4 ≈ 0.9659

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

≈ — rounded to 4 decimal places.

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

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

sin 75° = sin(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 y-axis — sine, y ≈ 0.9659.

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: sin 75° = (√6 + √2)/4

See Also

Recalculate with different values ↑

Sine of nearby angles

Anglesin
98°≈ 0.9903
100°≈ 0.9848
101°≈ 0.9816
105°(√6 + √2)/4
106°≈ 0.9613
108°√(10 + 2√5)/4
110°≈ 0.9397

FAQ

What is sin 105°?

sin 105° = (√6 + √2)/4 ≈ 0.9659

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 −0.9705 for sin 105?

The calculator is in radian mode (RAD): it takes sin 105 as the sine of 105 radians, sin 105 rad ≈ −0.9705. Switch it to degrees (DEG): sin 105° = (√6 + √2)/4 ≈ 0.9659. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also