How the Trigonometric Functions Calculator works
The six trigonometric functions describe the relationship between an angle and the coordinates of a point on the unit circle (a circle of radius 1 centered at the origin). For an angle θ measured counter-clockwise from the positive x-axis, the point where the angle's ray meets the unit circle has coordinates (cos θ, sin θ). Every other trig function is built from sine and cosine. This calculator takes an angle in degrees or radians and returns all six values.
Formula and method
Given an angle θ, the calculator first converts it to radians if needed, using radians = degrees × (π/180), since JavaScript's Math.sin and Math.cos (and every standard math library) expect radians. It then computes the two primary functions, sin θ and cos θ, directly. The remaining four functions are derived ratios: tan θ = sin θ / cos θ, csc θ = 1 / sin θ, sec θ = 1 / cos θ, and cot θ = cos θ / sin θ = 1 / tan θ. In a right triangle with an acute angle θ, these same functions equal opposite/hypotenuse (sine), adjacent/hypotenuse (cosine), and opposite/adjacent (tangent) — the unit-circle definition simply extends those triangle ratios to any angle, including negative angles and angles beyond 90°.
Common sources of error
- Degrees vs. radians: entering 90 when the calculator (or a formula) expects radians gives a completely different, meaningless result — always confirm which unit is selected.
- Undefined values: tan θ and sec θ are undefined whenever cos θ = 0 (at 90°, 270°, and every odd multiple of 90°); csc θ and cot θ are undefined whenever sin θ = 0 (at 0°, 180°, 360°, and every multiple of 180°).
- Sign errors by quadrant: sine, cosine, and tangent are positive or negative depending on which quadrant the angle falls in (remembered by "All Students Take Calculus" — All positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4).
Checking your result
Memorize a few exact benchmark values to sanity-check any output: sin 0° = 0, sin 30° = 0.5, sin 45° ≈ 0.7071, sin 60° ≈ 0.8660, sin 90° = 1, and cos θ follows the mirror pattern (cos 0° = 1, cos 90° = 0). You can also verify sin²θ + cos²θ = 1 for any angle, and confirm that tan θ = sin θ / cos θ matches the calculator's tangent output.
Applications
Trigonometric functions describe periodic motion (sound waves, AC electrical current, pendulum swings), are used to resolve forces and velocities into components in physics and engineering, drive navigation and surveying calculations (bearings, triangulation, GPS), and underlie rotation and lighting math in computer graphics and game development.