Trig calculator

Pick a function, choose degrees or radians, and type the angle. In radians you can type a decimal, like 2.5, or a multiple of π, like 3π/4 or pi/6. A ° after a number always means degrees, whichever unit is picked.

For any multiple of 15° the answer is exact, written with square roots, such as cos 150° = −√3/2, with the decimal alongside. Other angles get a decimal to six places. The six functions of the angle are all listed, so one angle gives you sin, cos, tan, csc, sec and cot together.

For arcsin, arccos and arctan, type the value instead: a decimal like 0.5, or an exact value like 1/2, √3/2 or -√2/2. The answer is the principal value, with every other angle between 0° and 360° that has the same sine, cosine or tangent.

In a right triangle, sin is the opposite side over the hypotenuse, cos is the adjacent side over the hypotenuse, and tan is opposite over adjacent. That only works for angles under 90°.

The unit circle extends them to every angle. Go anticlockwise round a circle of radius 1 from the positive x axis by the angle θ. The point you reach is (cos θ, sin θ), and tan θ is sin θ divided by cos θ. The drawing beside the answer shows that point, with the cos and sin as the sides of the triangle under it.

csc, sec and cot are the reciprocals: csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ, which is cos θ/sin θ.

Every angle has a reference angle, the angle between its radius and the nearest part of the x axis. In quadrant II it is 180° minus the angle, in quadrant III the angle minus 180°, and in quadrant IV 360° minus the angle. The size of each function is the same as at the reference angle. Only the sign changes.

The sign comes from the quadrant. cos is the x coordinate, so it is positive on the right half of the circle. sin is the y coordinate, positive on the top half. tan is their ratio, positive when they have the same sign. Starting top right and going anticlockwise, the functions that are positive are: all of them, then sin, then tan, then cos. The drawing letters the quadrants A, S, T and C to match.

So cos 150° has reference angle 30° and is in quadrant II, where cos is negative: cos 150° = −cos 30° = −√3/2.

tan θ = sin θ / cos θ, and at 90° and 270° cos θ is 0. Dividing by zero has no answer, so tan 90° and tan 270° are undefined, and so is sec, which is 1/cos. csc and cot divide by sin instead, so they are undefined at 0°, 180° and 360°.

On the graph these angles are vertical asymptotes, drawn dashed. The curve shoots up on one side and comes back from below on the other, and never touches the line.

arcsin x is the angle whose sine is x. But sin 30° and sin 150° are both 1/2, and so is sin 390°, so 'the angle' needs a rule. Each inverse gives one answer in a fixed range, the principal value: arcsin from −90° to 90°, arccos from 0° to 180°, and arctan between −90° and 90°.

The other answers follow from the circle. sin θ = sin(180° − θ), so the second sine solution is 180° minus the first. cos θ = cos(−θ), so the second cosine solution is 360° minus the first. tan repeats every 180°, so add 180°. Adding any whole number of full turns, 360° or 2π, gives more.

arcsin and arccos only take values from −1 to 1, because sine and cosine never leave that range. arcsin 2 has no answer among the real angles. arctan takes any number.

sin⁻¹ x is another way of writing arcsin x. It does not mean 1/sin x, which is csc x.

The sine graph is the height of the point on the unit circle as the angle grows, unrolled along the x axis. The cosine graph is the same wave moved 90° to the left. Both repeat every 360°, or 2π, and stay between −1 and 1.

The tangent graph repeats every 180°, or π, and has an asymptote wherever cos is 0. Secant and cosecant have the same asymptotes as tangent and cotangent, and never come between −1 and 1.

The sliders change y = A sin(Bx). A is the amplitude, the height of the wave above the middle line. The period is how far the wave goes before it repeats, 360° divided by B. Halve the period and the wave is squeezed to twice as many cycles.