Verify cofunction identities: sin(θ) = cos(90°−θ), tan(θ) = cot(90°−θ), sec(θ) = csc(90°−θ). Interactive unit circle & complementary angle analysis.
| Function f(θ) | Value f(θ) | Cofunction Identity | Complementary function f(90°−θ) | Status |
|---|
Current angle: 30.00°
Complement: 60.00°
In a right triangle, cofunctions link acute angles: sin A = cos B when A + B = 90°. On the unit circle, the x-coordinate of angle θ equals the y-coordinate of its complement (π/2 − θ).
Cofunction identities are fundamental trigonometric relationships that link the values of trigonometric functions of an angle θ and its complementary angle (90° − θ in degrees or π/2 − θ in radians). For any acute angle, these identities arise directly from the geometry of right triangles: the sine of an angle equals the cosine of its complement, the tangent equals the cotangent of its complement, and the secant equals the cosecant of its complement.
sin(θ) = cos(90° − θ) | cos(θ) = sin(90° − θ)
tan(θ) = cot(90° − θ) | cot(θ) = tan(90° − θ)
sec(θ) = csc(90° − θ) | csc(θ) = sec(90° − θ)
These identities hold for all real θ where both sides are defined. They are crucial for simplifying trigonometric expressions, solving equations, integrating transcendental functions, and even in physics (wave mechanics, optics). Euler’s formula provides an analytic proof: e^{iθ}=cosθ + i sinθ, and shifting by π/2 yields cofunction relations.
The concept of complementary angles and cofunctions dates back to Hellenistic mathematics, notably Ptolemy’s Almagest. In modern curricula, cofunction identities are essential for understanding the relationship between trigonometric graphs (phase shifts), and they appear in the Fourier transform, signal processing, and even architectural design. Our calculator not only computes numeric values but also reinforces the geometric meaning using the unit circle — a cornerstone of trigonometry.
Two sinusoidal waves with phases differing by 90° are in quadrature: sin(ωt + φ) = cos(ωt + φ − 90°). Cofunction identities help derive analytical expressions for resultant amplitude in interference patterns. Using this calculator, engineers can quickly verify phase shifts and cofunction equivalencies without manual error.
Consider a right triangle with angles A and B where A + B = 90°. For angle A, sin A = opposite/hypotenuse. For the complementary angle B, cos B = adjacent/hypotenuse. Since the side opposite A is adjacent to B, sin A = cos B. Hence sin A = cos(90° − A). Extending to unit circle definitions (x = cos θ, y = sin θ), the point for θ and π/2 − θ are symmetric across the line y = x, directly proving x-coordinate of one is y-coordinate of the other.
Using periodicity and symmetry, cofunction identities hold for any angle (including negative and >90°). For instance, sin(120°) = cos(120°−90°) = cos(30°). Our calculator automatically handles any real input, wrapping angles within the standard range for visualization, yet preserving identity accuracy via radian conversion.