Quadrants

Learning Objectives Determine the sign (positive or negative) of any trigonometric function in any of the four quadrants. Calculate the reference angle for any given angle in degrees or radians. Evaluate the six trigonometric functions for an angle given a point on its terminal side. Find the values of all trigonometric functions for an angle, given the value of one function and the quadrant. Evaluate trigonometric functions for angles greater than 360° (2π) or less than 0° by first finding a coterminal angle. Apply the CAST rule to quickly determine the sign of trigonometric ratios in each quadrant. Ever wonder how GPS systems pinpoint your location or how engineers model sound waves? 🛰️ It all comes back to understanding angles in different orientations! This tutorial div...

TermDefinitionExample Standard PositionAn angle is in standard position if its vertex is at the origin (0,0) of the Cartesian plane and its initial side lies along the positive x-axis.An angle of 120° is in standard position when its starting side is on the positive x-axis and it rotates 120° counter-clockwise. Terminal SideThe ray where the measurement of an angle ends. The quadrant in which the terminal side lies determines the signs of the trigonometric functions.For a 210° angle in standard position, the terminal side lies in Quadrant III. QuadrantsThe four regions into which the Cartesian plane is divided by the x-axis and y-axis. They are numbered I, II, III, and IV, starting from the top right and moving counter-clockwise.Quadrant I: x > 0, y > 0. Quadrant II: x < 0, y &gt...

The CAST Rule Quadrant I: All positive. Quadrant II: Sine positive. Quadrant III: Tangent positive. Quadrant IV: Cosine positive. A mnemonic (ASTC or CAST) to remember which trigonometric functions are positive in each quadrant. Start in Q-IV and spell CAST counter-clockwise, or start in Q-I and spell ASTC. 'A' for All, 'S' for Sine, 'T' for Tangent, 'C' for Cosine. The reciprocal functions (csc, cot, sec) follow the sign of their base function (sin, tan, cos). Reference Angle Formulas Q-I: θ' = θ Q-II: θ' = 180° - θ (or π - θ) Q-III: θ' = θ - 180° (or θ - π) Q-IV: θ' = 360° - θ (or 2π - θ) These formulas are used to find the reference angle (θ') for a given angle (θ) in each quadrant. The value of a trig funct...