Trigonometry Calculator

Results

In quadrant II, sine and cosecant are positive; the other four are negative.

The working

Figure

How to Use the Trigonometry Calculator

1. 1. Pick the direction. The six trig functions of an angle takes an angle and returns values; An angle from a trig value runs the inverse functions arcsin, arccos, arctan, arccsc, arcsec and arccot.
2. 2. Enter the angle or value. Type an angle in degrees, or in radians as a multiple of π such as 5pi/6 so it stays exact. For the inverse functions, type a decimal, a fraction such as 1/2, or a radical such as √3/2.
3. 3. Read the result. Each function appears exactly (where the angle is a multiple of 15°) and as a decimal, with the quadrant, the reference angle and the coterminal angle between 0° and 360°. Undefined values, such as tan 90°, are marked.
4. 4. Follow the working. The reference-angle step and the sign for the quadrant are written out, and a unit-circle figure shows the cosine, sine and tangent as lengths.

The Six Trig Functions

For an acute angle θ in a right triangle, sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse and tan θ = opposite / adjacent (the mnemonic SOH-CAH-TOA). The unit circle extends these to every angle: the point at angle θ on the circle of radius 1 is (cos θ, sin θ), and tan θ = sin θ / cos θ. The other three are reciprocals.

Exact Values of the Special Angles

These come from two triangles. Half of an equilateral triangle of side 2 is a 30-60-90 triangle with sides 1, √3 and 2, and an isosceles right triangle with legs 1 has hypotenuse √2. The calculator also knows every other multiple of 15° — for example sin 15° = (√6 − √2)/4 — by the angle-sum formulas, and all the angles in every quadrant by symmetry.

Quadrants and Reference Angles

The reference angle is the acute angle between the terminal side and the x-axis. A function at θ has the same size as at its reference angle; the quadrant sets the sign. In quadrant I all six functions are positive, in II sine and cosecant, in III tangent and cotangent, and in IV cosine and secant. An angle on an axis has no quadrant.

1. sin 150°. Quadrant II, reference angle 180° − 150° = 30°, sine positive: sin 150° = 1/2.
2. tan 7π/6. 7π/6 is 210°, in quadrant III with reference angle 30°; tangent is positive there: tan 7π/6 = √3/3.
3. cos(−60°). −60° is coterminal with 300°, in quadrant IV with reference angle 60°; cosine is positive: cos(−60°) = 1/2.

Inverse Trig Functions

A trig function repeats every turn, so it has no true inverse; the inverse functions return the principal value from a fixed range. arcsin gives an angle from −90° to 90°, arccos from 0° to 180°, and arctan strictly between −90° and 90°. arccsc and arcsec use the ranges of arcsin and arccos with the undefined point removed, and arccot returns an angle strictly between 0° and 180°, the usual calculus convention.

Every solution follows from the principal value. If sin θ = 1/2, then arcsin gives 30°, and the other solution in one turn is 180° − 30° = 150°, so θ = 30° + 360°k or 150° + 360°k. For cosine the partner is 360° − θ, and for tangent θ + 180°. Likewise arccos(−√2/2) = 135° = 3π/4, and arctan(−√3) = −60° = −π/3.

Degrees or Radians

A full turn is 360° = 2π radians, so 180° = π radians; multiply degrees by π/180 to get radians. Most wrong answers from a handheld calculator come from the wrong angle mode. Here the unit is stated beside the angle: a plain number in radians mode is read as radians, so sin 30 in degrees mode is 1/2 but sin 30 in radians mode is about −0.988.

Frequently Asked Questions

What are sine, cosine and tangent?

In a right triangle, sine is the opposite side over the hypotenuse, cosine is the adjacent side over the hypotenuse, and tangent is the opposite side over the adjacent side. On the unit circle, cosine and sine are the x- and y-coordinates of the point at the angle, and tangent is their ratio.

How do you find the exact value of sin 60°?

Cut an equilateral triangle of side 2 in half. The height is √3, the hypotenuse is 2, so sin 60° = √3/2. The same triangle gives cos 60° = 1/2 and tan 60° = √3.

What is a reference angle?

The acute angle that the terminal side of an angle makes with the x-axis: θ in quadrant I, 180° − θ in II, θ − 180° in III and 360° − θ in IV. It lets you reduce any angle to a first-quadrant value and then apply the quadrant's sign.

Why does arcsin give only one answer when there are many?

Because an inverse function has to return one value, so the output is restricted to a range where the function does not repeat: −90° to 90° for arcsin. The other solutions come from symmetry — 180° minus the angle for sine — plus whole turns of 360°.

Why is tan 90° undefined?

Because tan θ = sin θ / cos θ and cos 90° = 0, so the quotient divides by zero. The values of tan θ grow without bound as θ approaches 90°, and csc and cot are undefined at 0° and 180° for the same reason.

How do you find cosecant, secant and cotangent?

They are reciprocals: csc θ = 1 / sin θ, sec θ = 1 / cos θ and cot θ = 1 / tan θ = cos θ / sin θ. Find the sine, cosine or tangent first, then take the reciprocal; at 150°, sin is 1/2, so csc 150° = 2.

How do I convert radians to degrees?

Multiply by 180/π: π/6 rad is 30°, π/2 is 90° and 2π is 360°. Going the other way, multiply degrees by π/180. The calculator accepts radians as multiples of π, such as 5pi/6, and keeps them exact.