Trigonometric Functions Calculator

Trigonometric functions connect angles with ratios and coordinates. On the unit circle, cosine is the horizontal coordinate of the point reached by an angle and sine is its vertical coordinate. This interpretation works for positive angles, negative angles, rotations beyond one turn, and angles in every quadrant. It extends the familiar right-triangle definitions beyond acute angles while preserving the same relationships. Sine, cosine, and tangent are the three primary functions. In a right triangle relative to a chosen acute angle, sine is opposite divided by hypotenuse, cosine is adjacent divided by hypotenuse, and tangent is opposite divided by adjacent. Tangent also equals sine divided by cosine. Cosecant, secant, and cotangent are their reciprocals: csc θ = 1 / sin θ, sec θ = 1 / cos θ, and cot θ = cos θ / sin θ. Angles can be measured in degrees or radians. One complete turn is 360 degrees or 2π radians, so radians equal degrees multiplied by π / 180. Most programming-language trigonometric functions expect radians. When Degrees is selected, this calculator performs that conversion before evaluating the six functions. When Radians is selected, it uses the entered value directly. Choosing the correct unit is essential because 45 radians is very different from 45 degrees. Some trigonometric values are undefined. Tangent and secant divide by cosine, so they are undefined when cosine is zero, such as at 90 degrees plus whole half-turns. Cotangent and cosecant divide by sine, making them undefined at 0 degrees and whole half-turns. Computers approximate π and function values, which can leave an extremely small nonzero denominator at an axis angle. The calculator treats resulting enormous values as undefined rather than displaying a misleading finite number. The six outputs support geometry, wave analysis, periodic motion, navigation, surveying, physics, and engineering. Decimal results are rounded to six places for readability. For proofs and exact-value exercises, keep standard results such as sin 30° = 1/2 or cos 45° = √2/2 in symbolic form, using the decimal calculator to check numerical work and explore unfamiliar angles.