Tan 690 Degrees

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

What is tan 690°?

Answer

tan 690° =
−√3/3 ≈ −0.5774

Angle in radians: 690° = 23π/6. 690° = 360° + 330° — a full turn plus 330°, quadrant IV, where tangent is negative.

≈ — rounded to 4 decimal places.

tan 690° = tan(690° − 360°) = tan 330°

tan 330° = tan(360° − 30°) = −tan 30° = −√3/3

tan 690° = sin 690° / cos 690° = (−1/2) / (√3/2) = −√3/3

Half of an equilateral triangle with side 1: the hypotenuse is 1 and the leg opposite 30° is 1/2.

tan 30° = (1/2) / (√3/2) = 1/√3 = √3/3

xyOP

P(cos 690°, sin 690°) ≈ (0.8660, −0.5)

T(1, tan 690°) ≈ (1, −0.5774) — where line OP meets the line x = 1.

+1 turn: the drawing shows 330°.

All four functions of 690°
FunctionExact valueDecimal
sin 690°−1/2−0.5
cos 690°√3/2≈ 0.8660
tan 690°−√3/3≈ −0.5774
cot 690°−√3≈ −1.7321

Reference angle: tan 30° = √3/3

See Also

Recalculate with different values ↑

Tangent of nearby angles

Angletan
390°√3/3
400°≈ 0.8391
450°undefined
540°0
570°√3/3
690°−√3/3
720°0

FAQ

What is tan 690°?

tan 690° = −√3/3 ≈ −0.5774

How to remember sin 30°, sin 45° and sin 60°?

Write them as √1/2, √2/2, √3/2: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2. Cosines are the same numbers in reverse order: cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2; tan = sin / cos and cot = cos / sin. Here: tan 690° = −√3/3 ≈ −0.5774.

Why does a calculator show −2.2368 for tan 690?

The calculator is in radian mode (RAD): it takes tan 690 as the tangent of 690 radians, tan 690 rad ≈ −2.2368. Switch it to degrees (DEG): tan 690° = −√3/3 ≈ −0.5774. In radians the value is never undefined: π is irrational, so a whole number of radians never lands exactly on an axis.

See Also