Trig calculator: sin, cos, tan and their inverses

How to find sin, cos and tan of an angle

On a unit circle, the point at angle θ has coordinates (cos θ, sin θ), and tan θ = sin θ/cos θ. So sin 30° = 1/2, cos 180° = −1 and tan 45° = 1. Check the angle's unit first: sin 30 in radians is −0.988.

Degrees, radians and gradians

One full turn is 360° = 2π rad = 400 grad, so multiply degrees by π/180 to get radians. These three inputs describe the same 30° angle and give the same values; 100/3 is an exact number of gradians.

The same number in a different unit is a different angle, so a bare "cos 1" or "sin 30" has two answers. The unit is set separately, degrees unless changed, and never inferred from the number.

Calculus and most physics formulas work in radians; triangle and surveying problems usually give degrees. The OpenStax unit-circle lesson derives the coordinates behind these values.

Values of sin, cos and tan for common angles

All angles in this chart are in degrees. Decimals are rounded to ten places, and "undefined" marks a division by zero.

The exact values come from the 30-60-90 and 45-45-90 triangles: sin 30° = 1/2, sin 45° = √2/2 and sin 60° = √3/2, with cosine running the same values in reverse. Fifteen-degree steps add forms such as sin 15° = (√6 − √2)/4 and tan 15° = 2 − √3.

Signs follow the quadrant. Between 90° and 180° sine is positive and cosine negative, so cos 120° = −1/2; between 180° and 270° both are negative.

tan 90° is undefined because cos 90° = 0. Just below it the value grows without limit, tan 89° ≈ 57.29, but 90° itself has no tangent.

Angles above 360° and negative angles

Sine and cosine repeat every 360°, and tangent and cotangent every 180°. So sin 855° = sin 135° = √2/2 ≈ 0.7071067812, and cot 570° = cot 30° = √3 ≈ 1.7320508076.

Sine is an odd function and cosine an even one: sin(−30°) = −1/2, while cos(−30°) = cos 30° ≈ 0.8660254038. The reference angle is the acute angle to the x-axis, 45° for both 135° and 855°.

Is sin 37° equal to 3/5?

No. In a 3-4-5 right triangle the angle with sine 3/5 and cosine 4/5 is about 36.869898°, which physics problems round to 37°. At exactly 37°, sin ≈ 0.6018150232 and cos ≈ 0.7986355100: close to 0.6 and 0.8, but not equal.

Because 37° and 53° add to 90°, the sine of one is the cosine of the other. Some angles off the 15° grid also have exact forms: sin 18° = (√5 − 1)/4 and tan 67.5° = 1 + √2.

Secant, cosecant and cotangent

The reciprocal functions are sec θ = 1/cos θ, csc θ = 1/sin θ and cot θ = cos θ/sin θ (NIST DLMF §4.14).

Tangent and secant are undefined wherever cos θ = 0, at 90° and 270°; cosecant and cotangent wherever sin θ = 0, at 0° and 180°. So cot 90° is exactly 0, while cot 0° is undefined. OpenStax covers these restrictions.

Inverse trig functions: arcsin, arccos and arctan

An inverse function returns the angle with a given value: arcsin 0.5 = 30°. The notation sin⁻¹ x means arcsin x, not 1/sin x, which is the cosecant.

Each inverse returns one principal angle: arcsin in [−90°, 90°], arccos in [0°, 180°] and arctan in (−90°, 90°), following NIST DLMF §4.23.

Arccot here uses the continuous range (0°, 180°), which differs from DLMF's arctan(1/x) for negative x, so arccot(−1) is 135°, not −45°.

Other angles share the same value. sin θ = 0.5 for θ = 30° + 360°k or 150° + 360°k, and tan θ = 1 for θ = 45° + 180°k, where k is any integer.

The principal range also means arccos does not always undo cos: cos 240° = −0.5, but arccos(−0.5) = 120°. Arcsin and arccos accept values from −1 to 1, and arcsec and arccsc need x ≤ −1 or x ≥ 1.

Common mistakes

• The wrong angle unit. sin 30 is 0.5 in degrees and about −0.988 in radians, and pi/6 typed with degrees selected means 0.5236°, whose sine is about 0.009138.
• Reading sin⁻¹ as a reciprocal. sin⁻¹ 0.5 = 30°, but 1/sin 30° = csc 30° = 2.
• Writing a huge number for tan 90°. It is undefined, not large.
• Treating 37° as exactly 3/5. The 3-4-5 angle is 36.87°.
• Stopping at one inverse answer. 150° has the same sine as 30°.

Angle formats

• Units: degrees, radians or gradians; inverse results come out in the unit selected.
• Expressions: pi/4 in radians, 100/3 in gradians, and mixed numbers such as 37+1/2 for 37½°.
• Exact forms are given at multiples of 15°, such as √3/2 or 2 − √3.

For sides and angles of a triangle, use the right triangle calculator or the triangle solver. To convert an angle written in degrees, minutes and seconds, use the angle converter first.