Trigonometry Calculator

The six trigonometric functions

Trigonometry links the angles of a right triangle to the ratios of its sides. Label the side facing the angle the opposite, the side touching the angle (other than the longest one) the adjacent, and the longest side the hypotenuse. The three basic functions are then simple ratios.

sin(x) = opposite ÷ hypotenuse

cos(x) = adjacent ÷ hypotenuse

tan(x) = opposite ÷ adjacent = sin(x) ÷ cos(x)

Using the unit circle, the same functions are defined for every angle, positive or negative, not just those below 90°. The point at angle x on a circle of radius 1 has coordinates (cos x, sin x), and the tangent is the slope of the line from the centre to that point.

Sine and cosine always stay between −1 and 1 and repeat every 360° (2π radians). The tangent can take any value and repeats every 180° (π radians), with gaps at 90°, 270° and every angle 180° from those. For single-function pages see the sine calculator, the cosine calculator and the tangent calculator.

Exact values for common angles

Five angles come up so often that their sine, cosine and tangent are worth knowing by heart. The decimal columns are what the calculator shows.

To move between units, use radians = degrees × π ÷ 180. A half turn is 180° = π ≈ 3.141593 rad and a full turn is 360° = 2π ≈ 6.283185 rad.

How to use the calculator

1. Pick a function from the Function list.
2. Enter the number in the Number (x=) box. For sin, cos, tan, csc, sec and cot this is an angle; for the arc functions it is an ordinary real number.
3. Select degrees or radians under Metric. This only describes the angle you typed, so it has no effect on the arc functions.
4. Press Calculate. Direct functions give one line, rounded to at most six decimals. Arc functions give two lines, one in degrees and one in radians.

Example 1. sin(30°) = 0.50. Setting the metric to radians and entering 1 for cos gives cos(1) = 0.540302.

Example 2, a ramp. A 12 m ramp rises at 15°. The height gained is 12 × sin(15°) = 12 × 0.258819 = 3.11 m, and the horizontal run is 12 × cos(15°) = 12 × 0.965926 = 11.59 m.

Results are floating-point approximations. Values that are mathematically undefined, such as tan(90°), appear as a very large number (about 1.6 × 10¹⁶) because 90° cannot be stored as an exact number of radians.

Reciprocal functions: csc, sec and cot

Each basic function has a reciprocal partner. They are handy in calculus and in engineering formulas where dividing by a sine or cosine would otherwise clutter an expression.

csc(x) = 1 ÷ sin(x)

sec(x) = 1 ÷ cos(x)

cot(x) = 1 ÷ tan(x) = cos(x) ÷ sin(x)

Example: csc(30°) = 1 ÷ 0.5 = 2, sec(60°) = 1 ÷ 0.5 = 2 and cot(45°) = 1 ÷ 1 = 1. Because each one divides by a function that can be zero, csc is undefined at 0° and 180°, sec at 90° and 270°, and cot at 0° and 180°. In this calculator csc(0) is refused with a message and cot(0) shows inf.

For the cotangent on its own, see the cotangent calculator.

Inverse functions: finding an angle

The inverse (arc) functions go the other way: given a ratio, they return the angle. They are written arcsin, arccos and so on, or sin⁻¹, cos⁻¹, and should not be confused with reciprocals: sin⁻¹(x) is not 1 ÷ sin(x).

Example 1. A 3-4-5 right triangle has an angle opposite the side of length 3. Its sine is 3 ÷ 5 = 0.6, so the angle is arcsin(0.6) = 36.8699°, and the other acute angle is arccos(0.6) = 53.1301°. They add up to 90° as they should.

Example 2. arcsin(0.3) = 17.457603°, which is 0.304693 radians. If the input is outside the valid range, for instance arccos(2), the calculator prints nan (not a number) in both lines.

The arccsc, arcsec and arccot values are found from the other three: arccsc(x) = arcsin(1 ÷ x), arcsec(x) = arccos(1 ÷ x) and arccot(x) = 90° − arctan(x). So arccot(−1) = 90° − (−45°) = 135°. Single-function pages are available for arcsin, arccos, arctan and arccot.

Identities that help you check an answer

• Pythagorean identity: sin²(x) + cos²(x) = 1 for every x.
• Related forms: 1 + tan²(x) = sec²(x) and 1 + cot²(x) = csc²(x).
• Complementary angles: sin(x) = cos(90° − x) and tan(x) = cot(90° − x).
• Negative angles: sin(−x) = −sin(x), cos(−x) = cos(x) and tan(−x) = −tan(x).
• Quadrants: in the second quadrant (90° to 180°) only sine is positive, in the third only tangent, and in the fourth only cosine. For example sin(120°) = 0.866025 but cos(120°) = −0.50.

A quick sanity check for any result: squaring the sine and cosine of the same angle and adding them must give 1. For x = 1 radian, 0.841471² + 0.540302² = 0.708073 + 0.291927 = 1.

Common mistakes

• Wrong angle unit. sin(30) is 0.50 in degrees but about −0.988032 in radians.
• Expecting the arc functions to follow the Metric setting. They always return both degrees and radians.
• Reading sin⁻¹ as a reciprocal. Use csc for 1 ÷ sin(x).
• Forgetting that an inverse function gives only one of the infinitely many angles with that ratio. For example, sin(150°) is also 0.5 but arcsin(0.5) returns 30°.