Tan 690 Degrees: Tangent of 690° on the Unit Circle

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

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°.

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