Trig Calculator
All six trig functions of one angle, in degrees or radians. For the standard angles — multiples of 30° and 45° — the answer is exact, like √3/2 rather than 0.8660, with the quadrant and reference angle shown so you can see where it comes from.
How it works
How the Trig Calculator works
the part a result on its own leaves out
What the six functions are
Sine, cosine and tangent compare the sides of a right triangle, or equivalently the coordinates of a point on the unit circle: cosine is the x-coordinate, sine the y-coordinate, and tangent is sine divided by cosine. Cosecant, secant and cotangent are simply their reciprocals.
Why the reference angle does most of the work
Every angle behaves like an acute angle between 0° and 90°, with a sign decided by the quadrant it lands in. 135° has the same sine as 45°, because its reference angle is 45°, and it is positive because sine is positive in the second quadrant.
When a value is undefined
tan 90° is undefined, not a very large number. At 90° the cosine is 0, and tangent divides by cosine. The same applies to secant at 90° and to cosecant and cotangent at 0° and 180°. The calculator says undefined instead of printing a huge decimal.
Common mistakes
Common mistakes with trig functions
the errors that actually show up, not every possible slip
Calculating in the wrong angle mode
sin(30) is 0.5 in degrees and −0.988 in radians. Check the unit before reading the answer; the angle 30 means two different angles in the two modes.
Forgetting the sign from the quadrant
cos 135° is −√2/2, not +√2/2. The reference angle 45° gives the size, and the quadrant gives the sign: cosine is negative in the second quadrant.
Treating an undefined value as zero or infinity
tan 90° does not exist. It is not 0, and it is not a number you can use in a further calculation, because the division behind it has a zero denominator.
Questions
Asked often enough to answer here
answers that change how you work
What is the exact value of sin 45°?
√2/2, about 0.7071. The calculator keeps exact values for every multiple of 30° and 45°, where they exist.
What is tan 90°?
Undefined. tan is sin divided by cos, and cos 90° is 0, so the division cannot be done. Tangent has a vertical asymptote at 90° rather than a value.
How do I enter radians?
Switch the unit to radians and type the angle, using pi for π: pi/4, 3pi/4 or 2pi. A plain decimal such as 1 also works, giving a decimal answer.
Why do some answers show only decimals?
Exact forms exist only for the standard angles, the multiples of 30° and 45°. For an angle like 17° there is no neat radical form, so the calculator gives decimals to 10 places.
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