Sine, Cosine and Tangent Calculator

1. Choose From an angle or From a right triangle.
2. For an angle, type the value and pick degrees or radians. You get all six ratios, the reference angle and the quadrant.
3. For a triangle, fill in any two sides and leave the third blank. You get the angle, the missing side and the three ratios.
4. Use Copy, Save or Print to keep the result.

The angle in triangle mode is the one between the adjacent side and the hypotenuse, across from the opposite side.

In a right triangle, each trig function compares two sides as seen from one angle. The side across from the angle is the opposite side. The leg touching it is the adjacent side. The longest side is the hypotenuse.

The other three functions are simply the flips of these. They show up in calculus and physics.

Because the ratios depend only on the angle, any two triangles with the same angles give the same sine, cosine and tangent.

Triangles only cover angles between 0 and 90 degrees. The unit circle extends trig to every angle. Draw a circle of radius 1 at the origin and turn by the angle from the positive x-axis. The point you land on gives cosine and sine.

The signs change by quadrant. A handy phrase is "All Students Take Calculus": all ratios are positive in Quadrant I, only sine in II, only tangent in III and only cosine in IV.

The reference angle is the acute angle to the nearest x-axis. It gives the size of each ratio, and the quadrant gives the sign.

These five angles have exact values that are worth knowing by heart. They come from the 30-60-90 and 45-45-90 triangles.

As decimals, sqrt(3)/2 is about 0.866025, sqrt(2)/2 is about 0.707107 and sqrt(3)/3 is about 0.57735.

A full turn is 360 degrees or 2 pi radians, so 180 degrees equals pi radians. The same number means very different angles in each unit. That is why this calculator asks for the unit next to the angle.

For conversions and a full explanation, use our degrees vs radians calculator.

Take a right triangle with an opposite side of 3 and an adjacent side of 4.

Step 1: The two known sides are opposite and adjacent, so use TOA. tan(theta) = 3 / 4 = 0.75.

Step 2: Undo the tangent with arctan.

Step 3: The hypotenuse is sqrt(3 x 3 + 4 x 4) = 5. So sin(theta) = 0.6 and cos(theta) = 0.8. The other acute angle is 53.130102 degrees.

1. Using the Wrong DEG/RAD Mode

This is the most common error. sin(30) in radian mode is about -0.988, not 0.5. Always check the unit before you trust the answer.

2. Expecting a Number for tan 90 Degrees

Tangent is sine over cosine, and cos 90 degrees is 0. The value is undefined, not a huge number. The same goes for sec 90, cot 0 and csc 0 degrees.

3. Forgetting the Inverse Range

Inverse functions return only one angle. arcsin and arctan give answers from -90 to 90 degrees, and arccos gives 0 to 180. Another angle may share the same ratio.

4. Swapping Opposite and Adjacent

These names depend on which angle you use. The opposite side for one acute angle is the adjacent side for the other.

Results use standard double precision math and show up to 6 decimal places. Values within 1e-10 of 0, 1 or -1 are snapped, so sin 180 degrees shows 0 rather than a tiny leftover.

SOH-CAH-TOA is a memory aid for right triangles. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

Tangent equals sine divided by cosine. At 90 degrees the cosine is 0, and you cannot divide by zero, so tan 90 degrees has no value. The calculator shows undefined instead of a huge number.

It is probably in radian mode. In radians, sin(30) is about -0.988, because 30 radians is a very different angle from 30 degrees. Switch to degrees and sin 30 gives 0.5.

They are the reciprocal functions. Cosecant is 1 divided by sine, secant is 1 divided by cosine, and cotangent is 1 divided by tangent, which is the same as cosine over sine.

Use the inverse function that matches your two sides. With opposite and adjacent use arctan, with opposite and hypotenuse use arcsin, and with adjacent and hypotenuse use arccos. A 3-4-5 triangle gives about 36.87 degrees.

No, not for real angles. Sine and cosine are coordinates on a circle of radius 1, so they always fall between -1 and 1. Tangent can be any real number, while secant and cosecant are always at least 1 in size.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• NIST Digital Library of Mathematical Functions, section 4.14 (definitions and periodicity of the trig functions).
• OpenStax, Algebra and Trigonometry 2e (right triangle trig, the unit circle and special angles).
• Paul's Online Notes, Lamar University: Trig Function Evaluation (unit circle values and common errors).