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    Sin, Cos, Tan, Cot of an Angle

    Exact value in square roots and the decimal, with reduction to an acute angle on the unit circle

    Sin, Cos, Tan, Cot of an Angle

    Function
    Angle (°)

    Values of Sine, Cosine, Tangent and Cotangent

    On the unit circle the point of angle α is (cos α, sin α); tan α = sin α / cos α, cot α = cos α / sin α.
    Exact values from 0° to 360°
    Anglesincostancot
    0°010—
    30°1/2√3/2√3/3√3
    45°√2/2√2/211
    60°√3/21/2√3√3/3
    90°10—0
    120°√3/2−1/2−√3−√3/3
    135°√2/2−√2/2−1−1
    150°1/2−√3/2−√3/3−√3
    180°0−10—
    270°−10—0
    360°010—

    — — the value is undefined (division by zero)

    Example: sin 30°

    sin 30° = 1/2 = 0.5

    P(cos 30°, sin 30°) = (√3/2, 1/2)

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

    sin 30° = 1/2

    xyOP

    P(cos 30°, sin 30°) ≈ (0.8660, 0.5)

    Highlighted: the projection on the y-axis — sine, y = 0.5.

    Common mistakes

    • Radian mode on a calculator: sin 30 gives −0.9880, the sine of 30 radians. Switch it to degrees (DEG): sin 30° = 1/2.
    • tan 90° and cot 0° do not exist: cos 90° = 0 and sin 0° = 0, and the ratio would divide by zero.
    • The sign comes from the quadrant, not from the reference angle: sin 150° = 1/2, but sin 210° = −1/2 — both reduce to 30°.

    See Also

    Last Results

    Cos 215 Degrees: Cosine of 215° on the Unit Circle
    Cos 48 Degrees: Cosine of 48° on the Unit Circle
    Cos 46 Degrees: Cosine of 46° on the Unit Circle
    Cos 66 Degrees: Cosine of 66° on the Unit Circle
    Cos 11 Degrees: Cosine of 11° on the Unit Circle
    Cot 15 Degrees: Cotangent of 15° on the Unit Circle
    Cos 165 Degrees: Cosine of 165° on the Unit Circle
    Cos 127 Degrees: Cosine of 127° on the Unit Circle
    Cos 44 Degrees: Cosine of 44° on the Unit Circle
    Cos 12 Degrees: Cosine of 12° on the Unit Circle
    Cos 179 Degrees: Cosine of 179° on the Unit Circle
    Cos 71 Degrees: Cosine of 71° on the Unit Circle
    Cot 2 Degrees: Cotangent of 2° on the Unit Circle
    Cos 400 Degrees: Cosine of 400° on the Unit Circle
    Cos 150 Degrees: Cosine of 150° on the Unit Circle
    Cos 342 Degrees: Cosine of 342° on the Unit Circle
    Cos 64 Degrees: Cosine of 64° on the Unit Circle
    Cos 20 Degrees: Cosine of 20° on the Unit Circle
    Cos 130 Degrees: Cosine of 130° on the Unit Circle
    Cos 77 Degrees: Cosine of 77° on the Unit Circle
    Cot 13 Degrees: Cotangent of 13° on the Unit Circle
    Cot 164 Degrees: Cotangent of 164° on the Unit Circle
    Cos 200 Degrees: Cosine of 200° on the Unit Circle
    Cot 200 Degrees: Cotangent of 200° on the Unit Circle
    Cos 315 Degrees: Cosine of 315° on the Unit Circle
    Cos 6 Degrees: Cosine of 6° on the Unit Circle
    Cos 306 Degrees: Cosine of 306° on the Unit Circle
    Cos 23 Degrees: Cosine of 23° on the Unit Circle
    Cot 14 Degrees: Cotangent of 14° on the Unit Circle
    Cos 54 Degrees: Cosine of 54° on the Unit Circle
    Cot 198 Degrees: Cotangent of 198° on the Unit Circle
    Cos 85 Degrees: Cosine of 85° on the Unit Circle
    Cos 285 Degrees: Cosine of 285° on the Unit Circle
    Cos 160 Degrees: Cosine of 160° on the Unit Circle
    Cot 145 Degrees: Cotangent of 145° on the Unit Circle
    Cos 36 Degrees: Cosine of 36° on the Unit Circle
    Cos 47 Degrees: Cosine of 47° on the Unit Circle
    Cot 195 Degrees: Cotangent of 195° on the Unit Circle
    Cos 87 Degrees: Cosine of 87° on the Unit Circle
    Cot 127 Degrees: Cotangent of 127° on the Unit Circle
    Cos 16 Degrees: Cosine of 16° on the Unit Circle
    Cos 17 Degrees: Cosine of 17° on the Unit Circle
    Cos 159 Degrees: Cosine of 159° on the Unit Circle
    Cos 73 Degrees: Cosine of 73° on the Unit Circle
    Cos 74 Degrees: Cosine of 74° on the Unit Circle
    Cos 210 Degrees: Cosine of 210° on the Unit Circle
    Cos 300 Degrees: Cosine of 300° on the Unit Circle
    Cos 55 Degrees: Cosine of 55° on the Unit Circle
    Cot 101 Degrees: Cotangent of 101° on the Unit Circle
    Cos 35 Degrees: Cosine of 35° on the Unit Circle
    Cot 165 Degrees: Cotangent of 165° on the Unit Circle
    Cos 1 Degree: Cosine of 1° on the Unit Circle
    Cot 1 Degree: Cotangent of 1° on the Unit Circle
    Cos 167 Degrees: Cosine of 167° on the Unit Circle
    Cot 159 Degrees: Cotangent of 159° on the Unit Circle
    Cot 140 Degrees: Cotangent of 140° on the Unit Circle

    FAQ

    How to remember the table of values from 0° to 90°?

    Write the sines of 0°, 30°, 45°, 60°, 90° as √0/2, √1/2, √2/2, √3/2, √4/2 — that is 0, 1/2, √2/2, √3/2, 1. Cosines are the same numbers in reverse order. Then tan = sin / cos: tan 30° = √3/3, tan 45° = 1, tan 60° = √3.

    How to find the sine of an angle greater than 90°?

    Subtract full turns of 360°, then reduce to the acute reference angle through 180° or 360° — the function stays the same, the sign comes from the quadrant: sin 150° = sin(180° − 30°) = sin 30°, so sin 150° = 1/2 and cos 150° = −√3/2. Past 360°: sin 570° = −1/2.

    Why is tan 90° undefined?

    tan = sin / cos, and cos 90° = 0: you can't divide by zero. On the unit circle line OP of 90° runs parallel to the tangent line x = 1 and never meets it. The same holds for 270°, 450° and every 90° + 180°·k.