Mathematical Principles & Theorems
Trigonometric functions parameterized on the Cartesian unit circle \(x^2 + y^2 = 1\) with angle \(\theta\): \(x = \cos\theta, y = \sin\theta, \tan\theta = y/x\). Reciprocal functions: \(\sec\theta = 1/x, \csc\theta = 1/y, \cot\theta = x/y\). Pythagorean identities: \(\sin^2\theta + \cos^2\theta = 1, 1 + \tan^2\theta = \sec^2\theta, 1 + \cot^2\theta = \csc^2\theta\). Triangle laws: Law of Sines \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\); Law of Cosines \(c^2 = a^2 + b^2 - 2ab\cos C\).
Operating Instructions
- Drag the interactive unit circle point or enter precise angle \(\theta\) (in degrees or radians) into the numeric value box.
- Inspect the geometric line segments representing exact lengths of \(\sin, \cos, \tan, \sec, \csc, \cot\).
- Explore the Triangle Solver tab: enter side lengths or angles to compute unknown values via Sines/Cosines.
- Review quadrant signs and exact algebraic radical values for special angles (\(30^\circ, 45^\circ, 60^\circ, 90^\circ\)).