Coterminal and reference angles

Learning Objectives Define and identify angles in standard position, including their initial and terminal sides. Calculate positive and negative coterminal angles for a given angle in both degrees and radians. Determine the quadrant in which the terminal side of any given angle lies. Define and calculate the reference angle for any angle in all four quadrants, for both degrees and radians. Apply the concepts of coterminal and reference angles to evaluate the six trigonometric functions for angles outside the range [0°, 360°] or [0, 2π]. Connect the periodicity of trigonometric functions to the concept of coterminal angles. Ever wonder how a skateboarder can do a 1080° spin and land perfectly? 🛹 That's just three full rotations, ending in the exact same position as a 36...

TermDefinitionExample Angle in Standard PositionAn angle on the Cartesian plane whose vertex is at the origin and whose initial side lies on the positive x-axis. Counter-clockwise rotation is positive, and clockwise rotation is negative.An angle of 120° starts at the positive x-axis and rotates counter-clockwise into the second quadrant. Terminal SideThe ray where the measurement of an angle in standard position ends. The position of the terminal side determines the trigonometric function values.For a 120° angle, the terminal side is the ray in Quadrant II that forms a 120° angle with the positive x-axis. Coterminal AnglesAngles in standard position that have the same terminal side. They differ by integer multiples of a full rotation (360° or 2π radians).70°, 430° (70° + 360°), and -290°...

Finding Coterminal Angles In Degrees: \theta_{coterminal} = \theta + 360^{\circ}k \\ In Radians: \theta_{coterminal} = \theta + 2\pi k Use this formula to find angles that share the same terminal side as θ. 'k' can be any integer (positive, negative, or zero). To find the principal angle, add or subtract full rotations until the angle is between 0° and 360° (or 0 and 2π). Calculating Reference Angles (θ') Q1: \theta' = \theta \\ Q2: \theta' = 180^{\circ} - \theta \text{ or } \pi - \theta \\ Q3: \theta' = \theta - 180^{\circ} \text{ or } \theta - \pi \\ Q4: \theta' = 360^{\circ} - \theta \text{ or } 2\pi - \theta First, determine the quadrant of the angle's terminal side (using its principal angle). Then, apply the appropriate formula f...