Compute all six trigonometric ratios (sin, cos, tan, csc, sec, cot) for any angle, visualize on the unit circle, and evaluate inverse trigonometric functions. Ideal for homework, engineering, and interactive learning.
Compute the angle from a trigonometric ratio. Output in both degrees and radians (principal values).
Trigonometry studies relationships between angles and side lengths in triangles, but its modern definition revolves around the unit circle — a circle of radius 1 centered at the origin. For any angle θ measured from the positive x‑axis, the coordinates of the intersection point are (cos θ, sin θ). This definition extends the trigonometric functions to all real numbers and forms the foundation of periodic phenomena (sound, light, tides).
The earliest trigonometric tables were developed by Greek astronomers (Hipparchus) and later refined by Islamic scholars. The term “sine” derives from the Latin sinus. Euler introduced the modern notation and unified trigonometry with complex exponentials: e^{iθ} = cosθ + i sinθ — one of the most elegant formulas in mathematics. Our calculator lets you explore these relationships dynamically: change the angle and watch how sine corresponds to the vertical coordinate, cosine to the horizontal coordinate, and tangent to the slope.
Inverse functions (arcsin, arccos, arctan) return the angle whose sine, cosine, or tangent equals a given number. They are crucial for solving triangles, robotics (inverse kinematics), and GPS trilateration. The outputs are principal values: arcsin ∈ [-π/2, π/2], arccos ∈ [0, π], arctan ∈ (-π/2, π/2).
Electrical engineers use trigonometric identities to combine sinusoidal signals. For instance, when two sound waves with phase shift interact, the resultant amplitude is derived via cosine law: A = √(A₁² + A₂² + 2A₁A₂ cosΔφ). Our calculator helps quickly compute cosΔφ, evaluate resultant magnitude, and apply inverse cosine to determine phase difference — essential for noise cancellation algorithms.
Use the example buttons to explore special angles. Observe that sin²θ + cos²θ = 1 always holds, visible on the unit circle as the Pythagorean identity. The dynamic canvas also draws the sine segment (vertical line to the x-axis) to reinforce geometric meaning.