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BetterGrades Precalculus · Unit 9 · Unit map

Angles, Radians, and the Unit Circle

A radar arm begins on the positive x-axis, rotates 765 degrees counterclockwise, then 180 degrees clockwise. Where is its terminal side, and what is the smallest positive coterminal angle?

Precalculus · Unit 9

The lesson path

A radar arm begins on the positive x-axis, rotates 765 degrees counterclockwise, then 180 degrees clockwise. Where is its terminal side, and what is the smallest positive coterminal angle?

Core sequence

Angles, Radians, and the Unit Circle

Move in order when the material is new, or enter at the exact skill you need.

  1. 01Directed rotation and coterminal anglesRepresent rotations in standard position and generate all coterminal angles.
  2. 02Degree measure and angular coordinatesUse degrees, revolutions, and degree-minute-second notation to measure and compare rotation.
  3. 03Radian measure as normalized arc lengthDefine radian measure as signed arc length divided by radius and explain why it is independent of circle size.
  4. 04Arc length and sector areaDerive and use s=rs=r theta and A=(12)r2A=(\frac{1}{2})r^2 theta for angles measured in radians.
  5. 05Angular speed and linear speedRelate angular speed, radius, and tangential speed using v=rv=r omega.
  6. 06Wrapping the real line around the unit circleMap any real number tt to the unit-circle point reached by signed arc length tt.
  7. 07Sine and cosine as coordinate functionsDefine cosine and sine as the x- and y-coordinates of the unit-circle point P(t).
  8. 08Exact values from special trianglesDerive exact unit-circle values from 45−45−9045-45-90 and 30−60−9030-60-90 triangles.
  9. 09Reference angles, quadrants, and signsUse reference angles and quadrant signs to evaluate trigonometric functions of arbitrary rotations.
  10. 10Tangent and the reciprocal functionsDefine tangent, cotangent, secant, and cosecant from sine and cosine and state domains, signs, periods, and identities.
Source record

References used for this unit

  • Sundstrom & Schlicker, Trigonometry 1.1-1.6
  • Lippman & Rasmussen, Precalculus Vol. 2, 5.1-5.4
  • Yoshiwara, Trigonometry, Chapters 4 and 6
  • Corral, Trigonometry, Chapter 4
  • Stitz & Zeager, Precalculus, Chapter 10

The learner copy is original BetterGrades material informed by these rights-separated references; no long source passage is reproduced.