Find trigonometric ratios using reference angles

Learning Objectives Define a reference angle and its purpose. Determine the reference angle for any given angle in both degrees and radians. Identify the sign (+/-) of the six trigonometric ratios in each of the four quadrants using the ASTC (or CAST) rule. Calculate the exact value of trigonometric ratios for angles greater than 90° (π/2) by using reference angles and special triangles. Evaluate trigonometric ratios for negative angles and angles greater than 360° (2π) by first finding a positive coterminal angle. Apply the concept of reference angles to solve basic trigonometric equations within a specified domain. How can the cosine of 300° be exactly the same as the cosine of 60°? 🤔 Reference angles are the secret key that unlocks the cyclical nature of trigonometry! T...

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° in standard position starts at the positive x-axis and rotates counter-clockwise into the second quadrant. Reference Angle (θ_ref or α)The acute angle (always positive and between 0° and 90°) formed by the terminal arm of an angle in standard position and the horizontal x-axis.The reference angle for 150° is 180° - 150° = 30°. The reference angle for 5π/4 is 5π/4 - π = π/4. Terminal ArmThe ray of an angle in standard position that has been rotated from the initial side (the positive x-axis) to its final position.For a 210° angle, the terminal arm lies in the third quadrant. Coterminal Angles...

Reference Angle Formulas Let θ be the angle in standard position and θ_ref be its reference angle. Quadrant II: θ_ref = 180° - θ or θ_ref = π - θ Quadrant III: θ_ref = θ - 180° or θ_ref = θ - π Quadrant IV: θ_ref = 360° - θ or θ_ref = 2π - θ Use these formulas to find the acute reference angle based on the quadrant in which the terminal arm of the original angle θ lies. Note that angles in Quadrant I are their own reference angles. Trigonometric Value Procedure trig(θ) = (sign) * trig(θ_ref) The value of a trigonometric function for any angle θ is the value of that function for its reference angle θ_ref, prefixed with the correct positive or negative sign determined by the ASTC rule for the quadrant of θ.