Calculate all six trigonometric functions for an angle in degrees and identify its position in one rotation.
Complete the required fields and select the calculate button.
Angles are entered in degrees. A reciprocal or quotient is shown as undefined when its denominator is zero.
This trigonometry calculator evaluates sine, cosine, tangent, cotangent, secant, and cosecant for one angle in degrees. It combines the three primary functions and their reciprocals so their signs, domains, and relationships can be checked together.
The results are rounded to eight decimal places. At angles where a denominator is zero, the affected function is reported as undefined rather than as a very large numerical artifact.
Negative and coterminal angles are accepted. Subtracting or adding 360° leaves every displayed function unchanged, while tangent and cotangent also repeat after 180°.
sin θ = y/r; cos θ = x/r; tan θ = sin θ/cos θ
csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = cos θ/sin θ
The input conversion is radians = degrees × π/180. Functions with a zero denominator are undefined.
| Function | Right-triangle or reciprocal meaning |
|---|---|
| sin θ | Opposite/hypotenuse. |
| cos θ | Adjacent/hypotenuse. |
| tan θ | Opposite/adjacent. |
| csc θ | Reciprocal of sine. |
| sec θ | Reciprocal of cosine. |
| cot θ | Reciprocal of tangent, or cosine/sine. |
For θ = 30°, the page returns sin = 0.5, cos ≈ 0.8660254, tan ≈ 0.57735027, cot ≈ 1.73205081, sec ≈ 1.15470054, and csc = 2.
Multiplying each reciprocal pair gives approximately one. At 0°, cotangent and cosecant are undefined because sine is zero.
Sine and cosine always lie from −1 through 1 for real angles. Their reciprocals therefore have absolute value at least one wherever defined. Tangent and cotangent may take any finite real value.
Quadrants determine signs. In quadrant I all are positive; in quadrant II sine and cosecant are positive; in quadrant III tangent and cotangent are positive; in quadrant IV cosine and secant are positive.
Right-triangle side language is most direct for acute angles. The unit circle extends the same functions to negative angles and angles beyond 90°, where signed coordinates replace physical side lengths.
Near a zero denominator, a small angle change can produce a very large result. Measurement uncertainty and rounding therefore matter near 90° multiples for tangent and secant, and near 180° multiples for cotangent and cosecant.
Do not confuse reciprocal functions with inverse functions. Secant is one divided by cosine, whereas arccos returns an angle. Superscript minus-one notation can be ambiguous without context.
Eight decimals are display precision, not evidence that the entered angle is exact. Keep enough digits for a longer calculation, then round according to the source measurements.
Identity checks include sin²θ + cos²θ = 1, sec²θ = 1 + tan²θ, and csc²θ = 1 + cot²θ wherever all involved functions are defined.
Angles that differ by 360 degrees are coterminal, so all six values repeat. Sine, cosine, secant, and cosecant have fundamental period 360 degrees; tangent and cotangent repeat every 180 degrees. These periodic rules provide quick checks for large or negative inputs.
Special-angle exact forms such as √3/2 may be preferable in algebra, while the page displays decimals. A rounded decimal can verify magnitude and sign but should not replace an exact form when a problem explicitly requests one.
In applied work, confirm which angle a diagram defines and which side is opposite it. The same physical triangle has two acute angles, and exchanging them swaps sine with cosine and tangent with cotangent.
For coordinates on a circle of radius r, cosine supplies the horizontal coordinate divided by r and sine supplies the vertical coordinate divided by r. This picture explains both signs and the identity sin²θ + cos²θ = 1. It also shows why neither sine nor cosine can exceed one in absolute value.
Yes. Convert radians by multiplying by 180/π before entering the angle.
Tangent and secant divide by cosine; cotangent and cosecant divide by sine. A zero denominator has no finite reciprocal.
Secant is the reciprocal of cosine and returns a ratio. Arccos is an inverse function that returns a principal angle.
Displayed values are rounded. Using unrounded values gives one wherever both functions are defined.
Yes. Odd functions sine, tangent, cotangent, and cosecant change sign; even functions cosine and secant do not.
Check the quadrant signs, reciprocal pairs, and sin squared plus cos squared. Also compare with a familiar landmark angle.
The denominator is approaching zero. The function changes rapidly there and becomes undefined at the exact asymptote.