Find trigonometric ratios using the unit circle

Learning Objectives Define the unit circle and its relationship to trigonometric functions. Identify the coordinates of points on the unit circle corresponding to special angles (multiples of π/6, π/4, and π/2). Determine the six trigonometric ratios (sin, cos, tan, csc, sec, cot) for any angle given a point (x, y) on the unit circle. Use the unit circle to find trigonometric ratios for angles greater than 2π (360°) and negative angles by finding coterminal angles. Evaluate trigonometric functions of quadrantal angles (0, π/2, π, 3π/2). Relate the signs of trigonometric ratios to the quadrant in which the terminal side of the angle lies. Ever wondered how your favorite video game renders complex 3D rotations or how engineers model sound waves? 🎮 It all revolves around the e...

TermDefinitionExample Unit CircleA circle centered at the origin (0,0) of the Cartesian plane with a radius of exactly 1. Its equation is x² + y² = 1.The point (√2/2, √2/2) lies on the unit circle because (√2/2)² + (√2/2)² = 2/4 + 2/4 = 1. Standard PositionAn angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. Angles are measured counter-clockwise for positive values and clockwise for negative values.An angle of 120° starts at the positive x-axis and rotates counter-clockwise into the second quadrant. Terminal SideThe ray of an angle in standard position that has been rotated from the initial side. The point where the terminal side intersects the unit circle gives the coordinates (x, y) used for trigonometric ratios.For an an...

Unit Circle Trigonometric Definitions For an angle θ whose terminal side intersects the unit circle at point P(x, y): sin(θ) = y, cos(θ) = x, tan(θ) = y/x, csc(θ) = 1/y, sec(θ) = 1/x, cot(θ) = x/y These definitions are the foundation of unit circle trigonometry. The x-coordinate of the intersection point is the cosine of the angle, and the y-coordinate is the sine. All other ratios are derived from these two. Pythagorean Identity on the Unit Circle Since any point (x, y) on the unit circle satisfies x² + y² = 1, it follows that cos²(θ) + sin²(θ) = 1. This fundamental identity is a direct result of the Pythagorean theorem applied to the unit circle. It is essential for solving trigonometric equations and proving other identities. Signs by Quadrant (ASTC Rule) Q1: All po...