Degrees, Radians, and Relevant Unit-Circle Relationships

Objective: Convert angle measures with π radians equal to 180°.

Prerequisite skills: circle circumference and right-triangle ratios. A radian measures arc length relative to radius. A full circle has angle 2π radians, matching 360°. Multiply degrees by π/180 to convert to radians and radians by 180/π to convert back. Common error: using 360 in place of 180 in a one-π conversion. Restrictions and interpretation: A full revolution is 2π radians or 360°; unit-circle coordinates are (cosθ,sinθ) and quadrant signs matter.

Objective: Connect sine and cosine to horizontal and vertical coordinates.

A point at angle θ on the unit circle has coordinates (cosθ,sinθ). Its coordinate squares sum to 1. Useful first-quadrant coordinates are 30°: (√3/2,1/2), 45°: (√2/2,√2/2), and 60°: (1/2,√3/2). These come from special right triangles scaled to hypotenuse 1. At 0°,90°,180°,270°, the points are (1,0),(0,1),(−1,0),(0,−1). Apply quadrant signs to reference-angle values. Common error: swapping sine and cosine coordinates. Restrictions and interpretation: A full revolution is 2π radians or 360°; unit-circle coordinates are (cosθ,sinθ) and quadrant signs matter.

Objective: Use a reference angle and signs to find an exact trig value.

A reference angle relates an angle in another quadrant to an acute triangle. In quadrant II, x and cosine are negative while y and sine are positive. Adding full revolutions does not change unit-circle coordinates. Common error: keeping an acute cosine positive in quadrant II. Restrictions and interpretation: A full revolution is 2π radians or 360°; unit-circle coordinates are (cosθ,sinθ) and quadrant signs matter. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.