Sin 165 Degrees

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

What is sin 165°?

Answer

sin 165° =
(√6 − √2)/4 ≈ 0.2588

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

≈ — rounded to 4 decimal places.

sin 165° = sin(180° − 15°) = sin 15° = (√6 − √2)/4

P(cos 165°, sin 165°) = (−(√6 + √2)/4, (√6 − √2)/4)

sin 15° = sin(45° − 30°) = (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4

xyOP

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

Highlighted: the projection on the y-axis — sine, y ≈ 0.2588.

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: sin 15° = (√6 − √2)/4

See Also

Recalculate with different values ↑

Sine of nearby angles

Anglesin
160°≈ 0.3420
162°(√5 − 1)/4
164°≈ 0.2756
165°(√6 − √2)/4
167°≈ 0.2250
175°≈ 0.0872
179°≈ 0.0175

FAQ

What is sin 165°?

sin 165° = (√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 0.9978 for sin 165?

The calculator is in radian mode (RAD): it takes sin 165 as the sine of 165 radians, sin 165 rad ≈ 0.9978. Switch it to degrees (DEG): sin 165° = (√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