Foundations Summary

Trigonometry · Axiom Academy

SUMMARY Foundations of Trigonometry Let's review how circular motion connects angles to coordinates through the unit circle, forming the foundation for all trigonometric functions. Degrees: One complete rotation equals 360°. This divides circles into convenient fractions (90° for quarter turns, 180° for half rotations). Radians: Measure angles by arc length on the unit circle. One radian is the angle that creates an arc of length 1. A full circle is 2π radians. Conversion: 180° = π radians. Multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees. Coterminal Angles: Angles that share the same terminal side differ by multiples of 360° or 2π. For example, 30°, 390°, and -330° are coterminal. Definition: A circle with radius 1 centered at the origin. Every point satisfies x² + y² = 1 . Coordinates from Angles: An angle θ measured counterclockwise from the positive x-axis determines a unique point (x, y) on the circle. Special Angles: 0°, 30°, 45°, 60°, 90° (and their multiples) have exact coordinate values involving 0, 1, ½, √2/2, and √3/2. Symmetry: The unit circle has symmetry across both axes and through the origin, creating predictable patterns in all four quadrants. Reference Angle: The acute angle between the terminal side and the x-axis. Always between 0° and 90° (or 0 and π/2 radians). ASTC Rule: "All Students Take Calculus" tells us which functions are positive: All in QI, Sine in QII, Tangent in QIII, Cosine in QIV.

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