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 for key angles (e.g., 0, π/6, π/4, π/3, π/2 and their multiples). Find the six trigonometric ratios (sin, cos, tan, csc, sec, cot) for any angle whose terminal side intersects the unit circle at a known point (x, y). Determine the sign (+/-) of any trigonometric ratio by identifying the quadrant of the angle's terminal side. Evaluate trigonometric functions for angles greater than 360° (2π radians) and negative angles by using coterminal angles. Use reference angles to find the trigonometric ratios for non-quadrantal angles in any quadrant. Ever wondered how your phone's GPS or a video game's physics engine calculate...

TermDefinitionExample Unit CircleA circle with a radius of 1 centered at the origin (0,0) of the Cartesian plane. 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. Terminal SideFor an angle in standard position, it is the ray that has been rotated from the initial position on the positive x-axis.For a 120° angle, the terminal side lies in Quadrant II. Standard PositionAn angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.An angle of -90° in standard position has its terminal side on the negative y-axis. Reference AngleThe acute angle (always positive) formed by the terminal side of an angle θ and the horizontal x-axis.The reference angle for 150° is 180° - 150°...

Unit Circle Definitions of Trigonometric Ratios 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. This set of definitions is the foundation for finding trigonometric ratios on the unit circle. The x-coordinate is the cosine of the angle, and the y-coordinate is the sine. Pythagorean Identity on the Unit Circle cos²(θ) + sin²(θ) = 1 This is a direct result of applying the unit circle definitions (x = cos(θ), y = sin(θ)) to the equation of the unit circle (x² + y² = 1). It is one of the most fundamental trigonometric identities. Signs of Ratios by Quadrant (ASTC) Quadrant I: All positive. Quadrant II: Sine positive. Quadrant III: Tangent positive. Quadrant IV: Co...