Trigonometry Calculator

Put a circle of radius 1 (the unit circle) on a graph, centred at (0, 0). Start at the point (1, 0) and turn anticlockwise by the angle θ. The point you land on is (cos θ, sin θ): cosine is how far across, sine is how far up. Every other function comes from those two. Turning clockwise gives a negative angle, and going past 360° just goes round again. For an angle in a right triangle, the same definitions give the familiar SOH CAH TOA. To solve a whole triangle, use the triangle calculator or, for a right triangle, the Pythagorean theorem calculator.

The formula

Point on the unit circle at angle θ = (cos θ, sin θ)
Right triangle: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent
tan θ = sin θ ÷ cos θ (undefined when cos θ = 0)
csc θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, cot θ = 1 ÷ tan θ = cos θ ÷ sin θ
Radians = degrees × π ÷ 180. Gradians = degrees × 10 ÷ 9 (400 grad in a full turn)
Degrees from D° M′ S″ = D + M ÷ 60 + S ÷ 3600
Reference angle: θ in quadrant 1; 180° − θ in quadrant 2; θ − 180° in quadrant 3; 360° − θ in quadrant 4
sin⁻¹ gives −90° to 90°, cos⁻¹ gives 0° to 180°, tan⁻¹ gives between −90° and 90°
All answers: sin θ = x → θ = sin⁻¹ x + 360°k or 180° − sin⁻¹ x + 360°k; cos θ = x → θ = ±cos⁻¹ x + 360°k; tan θ = x → θ = tan⁻¹ x + 180°k

Worked example

θ = 30° (the starting value).

• In radians: 30 × π ÷ 180 = π/6 = 0.523599 rad. In gradians: 30 × 10 ÷ 9 = 33.333333 grad.
• 30° is between 0° and 90°, so it is in quadrant 1, every function is positive and the reference angle is 30° itself.
• The point on the unit circle is (√3/2, 1/2), so sin 30° = 1/2 = 0.5 and cos 30° = √3/2 = 0.866025.
• tan 30° = sin ÷ cos = 0.5 ÷ 0.866025 = 1/√3 = √3/3 = 0.57735.
• csc 30° = 1 ÷ 0.5 = 2. sec 30° = 1 ÷ 0.866025 = 2√3/3 = 1.154701. cot 30° = 1 ÷ 0.57735 = √3 = 1.732051.
• Check: sin² + cos² = 0.25 + 0.75 = 1.
• Going back: sin⁻¹(0.5) = 30°, and the other answer from 0° to 360° is 180° − 30° = 150°.

Exact values of the special angles

A handy pattern for sin at 0°, 30°, 45°, 60° and 90°: √0/2, √1/2, √2/2, √3/2, √4/2. Cosine runs the same list backwards. The calculator also knows the less common exact values:

Signs in each quadrant (ASTC)

To find any value: work out the reference angle, take the function of that, and give it the sign for the quadrant. sin 210° = −sin 30° = −1/2, because 210° is in quadrant 3, where sine is negative.

Inverse functions and every answer

sin⁻¹, cos⁻¹ and tan⁻¹ (also written arcsin, arccos and arctan) turn a value back into an angle. Because the functions repeat, each one picks a single answer, the principal value. To list every answer from 0° to 360°: for sine use θ and 180° − θ; for cosine use θ and 360° − θ; for tangent use θ and θ + 180°. Then add or take away 360° (180° for tan and cot) to bring them into the range you want. sin⁻¹ and cos⁻¹ only accept values from −1 to 1, while csc⁻¹ and sec⁻¹ only accept values of 1 or more, or −1 or less. In Advanced, cot⁻¹ follows the common convention of answers between 0° and 180°.

atan2(y, x)

tan⁻¹(y ÷ x) cannot tell the point (3, −4) from (−3, 4), since both give y ÷ x = −4/3. atan2 looks at the signs of x and y separately and gives the angle of the point from −180° to 180°. For y = 3 and x = −4, tan⁻¹(3 ÷ −4) = −36.87°, but the point is in quadrant 2, so atan2 adds 180°: 143.13°. Spreadsheets write it as ATAN2(x, y), with x first.

Hyperbolic functions and identities

In Advanced, sinh, cosh and tanh (and their reciprocals) are worked out for the angle in radians: sinh x = (eˣ − e⁻ˣ) ÷ 2 and cosh x = (eˣ + e⁻ˣ) ÷ 2. They describe a hanging chain or cable, not a circle. The identity check uses your numbers to confirm sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ, sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ.

The calculator works in double precision and rounds only for display. Values that should be exactly 0, 1 or 0.5 are cleaned of floating-point dust, and functions that divide by zero show “undefined”.