Cos 165 Degrees

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

What is cos 165°?

Answer

cos 165° =
−(√6 + √2)/4 ≈ −0.9659

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

≈ — rounded to 4 decimal places.

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

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

cos 15° = cos(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 x-axis — cosine, x ≈ −0.9659.

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: cos 15° = (√6 + √2)/4

See Also

Recalculate with different values ↑

Cosine of nearby angles

Anglecos
160°≈ −0.9397
162°−√(10 + 2√5)/4
164°≈ −0.9613
165°−(√6 + √2)/4
167°≈ −0.9744
175°≈ −0.9962
179°≈ −0.9998

FAQ

What is cos 165°?

cos 165° = −(√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 −0.0663 for cos 165?

The calculator is in radian mode (RAD): it takes cos 165 as the cosine of 165 radians, cos 165 rad ≈ −0.0663. Switch it to degrees (DEG): cos 165° = −(√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