Find trigonometric ratios using reference angles

Learning Objectives Define a reference angle for any angle in standard position. Determine the reference angle for a given angle in both degrees and radians. Identify the sign (+/-) of the six trigonometric functions in each of the four quadrants using the ASTC rule. Use reference angles to find the exact value of sine, cosine, and tangent for angles whose terminal side lies outside the first quadrant. Evaluate all six trigonometric functions (sin, cos, tan, csc, sec, cot) for any angle whose reference angle is a special angle (30°, 45°, 60° or π/6, π/4, π/3). Apply the concept of reference angles to evaluate trigonometric functions for angles greater than 360° (2π) or negative angles by first finding a coterminal angle. Ever wonder how your calculator instantly finds the si...

TermDefinitionExample Standard PositionAn angle is in standard position if its vertex is at the origin of a Cartesian coordinate system and its initial side coincides with the positive x-axis.An angle of 210° drawn with its vertex at (0,0) and starting on the positive x-axis is in standard position. Its terminal side is in Quadrant III. Reference Angle (θ')The acute, positive angle formed by the terminal side of a given angle (θ) and the horizontal x-axis. The reference angle is always between 0° and 90° (or 0 and π/2 radians).For an angle of 150°, the terminal side is in Quadrant II. The reference angle is the angle between the terminal side and the negative x-axis, which is 180° - 150° = 30°. Terminal SideThe ray of an angle in standard position that rotates from the initial side t...

Reference Angle Formulas For an angle θ: Quadrant II: θ' = 180° - θ or θ' = π - θ Quadrant III: θ' = θ - 180° or θ' = θ - π Quadrant IV: θ' = 360° - θ or θ' = 2π - θ Use these formulas to calculate the reference angle (θ') based on the quadrant in which the terminal side of the original angle (θ) lies. The ASTC Rule A mnemonic for determining the sign of trigonometric functions: Quadrant I (A): All functions are positive. Quadrant II (S): Sine (and csc) are positive. Quadrant III (T): Tangent (and cot) are positive. Quadrant IV (C): Cosine (and sec) are positive. After finding the reference angle, use this rule to determine whether the trigonometric ratio of the original angle is positive or negative based on its quadrant. A common mn...