The six functions and how they relate
On the unit circle, an angle θ lands at the point (cos θ, sin θ). The other four functions follow from those two: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ, sec θ = 1/cos θ and csc θ = 1/sin θ.
Whenever a denominator is zero the function is undefined. That is why tan 90° and sec 90° have no value, and why cot and csc are undefined at 0° and 180°.
Radians and degrees measure the same angle: radians = degrees × π/180. You can type an angle in radians as a decimal or with π, such as π/6 or 2π/3.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° (π/6) | 1/2 | √3/2 | √3/3 |
| 45° (π/4) | √2/2 | √2/2 | 1 |
| 60° (π/3) | √3/2 | 1/2 | √3 |
| 90° (π/2) | 1 | 0 | undefined |
Worked example: signs outside the first quadrant
For 210°, the reference angle is 210° − 180° = 30°, and 210° lies in quadrant III where sine and cosine are both negative. So sin 210° = −1/2, cos 210° = −√3/2 and tan 210° = √3/3, which is positive because the two negatives cancel.
Working backwards, acos(−0.5) = 120° = 2π/3. The inverse returns only the principal value; 240° also has a cosine of −0.5.
Principal values of the inverse functions
asin returns angles from −90° to 90°, acos from 0° to 180° and atan strictly between −90° and 90°. asin and acos only accept inputs from −1 to 1; acsc and asec only accept values of 1 or more in size.
Conventions for acot differ between textbooks. This calculator returns acot in the range 0° to 180°, so acot(−1) = 135° and acot(0) = 90°.
Graphs of the functions and their inverses
Graph mode plots any of the six functions, or sine and cosine together, over the range you choose, in degrees, radians or multiples of π. Sine, cosine, cosecant and secant repeat every 360° (2π); tangent and cotangent repeat every 180° (π).
Tangent and secant have vertical asymptotes where cos θ = 0 (90° + 180°k), and cotangent and cosecant where sin θ = 0 (180°k). The graph leaves out values larger than the clip limit, so the line breaks at each asymptote instead of joining across it. It also never connects two points on either side of one.
The inverse graphs show principal values: asin and acos exist only for −1 ≤ x ≤ 1, acsc and asec only for |x| ≥ 1, and atan and acot for every x. The table under the chart lists sample points you can copy or download.