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BetterGrades Precalculus · Unit 10 · Unit map

Trigonometric Functions and Periodic Models

A point starts at (1,0) and rotates counterclockwise one revolution in 8 seconds. Sketch its vertical coordinate against time and label zeros, extrema, and period.

Precalculus · Unit 10

The lesson path

A point starts at (1,0) and rotates counterclockwise one revolution in 8 seconds. Sketch its vertical coordinate against time and label zeros, extrema, and period.

Core sequence

Trigonometric Functions and Periodic Models

Move in order or enter at the exact skill you need.

  1. 01Building the sine graph from circular motionConstruct y=sinty=sin t from the vertical coordinate of a rotating unit-circle point.
  2. 02Building the cosine graph from circular motionConstruct y=costy=cos t from the horizontal coordinate of circular motion and compare it with sine.
  3. 03Amplitude, reflection, and midlineInterpret A and D in y=Ay=A sin x+Dx+D or y=Ay=A cos x+Dx+D.
  4. 04Period, frequency, and angular frequencyRelate inside scale B to period, ordinary frequency, and angular frequency.
  5. 05Phase shift and timingInterpret B(x-C) as a horizontal timing shift and recover C from graph or context.
  6. 06The general sinusoidal functionBuild and analyze y=Asin(B(x−C))+Dy=A sin(B(x-C))+D and equivalent cosine forms.
  7. 07Tangent and cotangent graphsDerive tangent and cotangent graphs from quotient definitions, zeros, asymptotes, and period pi.
  8. 08Secant and cosecant graphsConstruct secant and cosecant graphs as reciprocals of cosine and sine.
  9. 09Symmetry, periodicity, and the six-function familyCompare domains, ranges, parity, periods, zeros, and asymptotes of all six trig functions.
  10. 10Inverse trigonometric functions and branch restrictionsDefine inverse sine, cosine, and tangent using one-to-one branches and principal-value ranges.
  11. 11Basic trigonometric equationsSolve sin x=k,x=k, cos x=k,x=k, and tan x=kx=k using exact values, inverse functions, periodicity, and interval restrictions.
  12. 12Sinusoidal modeling and regressionFit and critique sinusoidal models using amplitude, midline, period, phase, residuals, and domain.
Assessment path

Review, practice, demonstrate, investigate

Source record

References used for this unit

  • Sundstrom & Schlicker, Trigonometry 2.1-2.6
  • Lippman & Rasmussen, Precalculus Vol. 2, Chapter 6
  • Yoshiwara, Trigonometry, Chapters 4, 6, and 7
  • AP Precalculus framework, Trigonometric and Polar Functions