How the Trig Function Calculator Works
This calculator evaluates all six trigonometric functions — sine, cosine, tangent, cosecant, secant, and cotangent — for any angle you enter, in either degrees or radians. Trigonometric functions relate an angle to ratios of the sides of a right triangle, or equivalently to the x- and y-coordinates of a point on the unit circle.
Definitions from the unit circle
For an angle θ measured from the positive x-axis, place a point on the unit circle (radius 1) at that angle. By definition, the point's coordinates are (cos θ, sin θ). Tangent is the ratio tan θ = sin θ / cos θ, which also equals the slope of the line from the origin to that point. In a right triangle, the same functions equal sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent.
Reciprocal functions and identities
Cosecant, secant, and cotangent are defined as reciprocals: csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ = cos θ/sin θ. Every angle satisfies the Pythagorean identity sin²θ + cos²θ = 1, from which 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ follow directly. Sine, cosine, cosecant, and secant repeat every 360° (2π radians); tangent and cotangent repeat every 180° (π radians) because reflecting the point to the opposite side of the circle leaves their ratio unchanged.
Undefined values and common mistakes
Tangent and secant are undefined wherever cos θ = 0 — at 90°, 270°, and any angle of the form 90° + 180°n — because both divide by cos θ. Cosecant and cotangent are undefined wherever sin θ = 0 — at 0°, 180°, and any multiple of 180° — because both divide by sin θ. A frequent mistake is mixing degree and radian mode: sin(30) evaluated as 30 radians (≈ -0.988) is not the same as sin(30°) (= 0.5), so always confirm which unit is selected before reading a result.