Trigonometry, polar geometry and precalculus representations

Lesson sequence

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1. Measure rotation and derive exact values

   Lesson details and objectives

   Connect arc length, signed rotation and special-triangle coordinates.

   • Convert degrees and radians with units.
   • Derive exact sine/cosine values and quadrant signs.
   • Distinguish coterminal angles from equal real numbers.

   6 practice tasks.

2. Build sine and cosine graphs from rotation

   Lesson details and objectives

   Map unit-circle coordinates to graphs and transform their periods and heights.

   • Read amplitude, midline, period and phase.
   • Map quarter-period landmarks.
   • Handle negative outside coefficients without negative amplitudes.

   6 practice tasks.

3. Track tangent and reciprocal-function domains

   Lesson details and objectives

   Distinguish ratios from reciprocals and locate zeros and poles.

   • Use tan, sec, csc and cot definitions.
   • Determine excluded angles and periods.
   • Explain asymptotic behavior with denominator signs.

   6 practice tasks.

4. Choose a principal inverse branch

   Lesson details and objectives

   Recover angles without confusing one principal answer with every possible angle.

   • Use principal ranges of arcsin, arccos and arctan.
   • Evaluate inverse compositions with restrictions.
   • Distinguish inverse functions from reciprocal functions.

   6 practice tasks.

5. Prove identities on their shared domains

   Lesson details and objectives

   Use definitions and algebra to justify all-input equalities rather than isolated coincidences.

   • Derive Pythagorean identities by valid division.
   • Preserve restrictions when simplifying.
   • Use counterexamples to refute false identities.

   6 practice tasks.

6. Derive and use addition and double-angle formulas

   Lesson details and objectives

   Connect angle addition to rotations and retain both cross terms.

   • Apply sine/cosine sum and difference identities.
   • Derive double-angle forms.
   • Handle tangent-addition denominator failures.

   6 practice tasks.

7. Find complete interval and general solutions

   Lesson details and objectives

   Combine factoring, inverse branches and periodicity without discarding roots.

   • Solve factored trigonometric equations.
   • State general solutions with integer parameters.
   • Filter by a requested interval and original domain.

   6 practice tasks.

8. Fit a periodic model and test its assumptions

   Lesson details and objectives

   Translate extrema and timing into a sinusoid and distinguish fitting from evidence.

   • Determine midline, amplitude, period and phase from context.
   • Predict using units and landmarks.
   • Identify nonunique fits and model limitations.

   6 practice tasks.

9. Use the sine law without losing a triangle

   Lesson details and objectives

   Compare side-angle pairs and enumerate zero, one or two SSA triangles.

   • Apply a/sin A=b/sin B to nondegenerate triangles.
   • Check supplementary inverse-sine candidates.
   • Classify triangle feasibility before reporting a solution.

   6 practice tasks.

10. Solve triangles from sides or an included angle

    Lesson details and objectives

    Derive the cosine law from a coordinate projection and use it with triangle conditions.

    • Apply the cosine law for SAS and SSS data.
    • Classify angles by squared-side comparisons.
    • Use an included-angle area formula and reject degenerate data.

    6 practice tasks.

11. Measure vector alignment and projection

    Lesson details and objectives

    Separate vectors from points and connect the dot product to angles and work.

    • Compute magnitude, dot product and projection.
    • Interpret orthogonality and signed alignment.
    • State zero-vector and unit restrictions.

    6 practice tasks.

12. Represent a point with radius and angle

    Lesson details and objectives

    Convert coordinates while preserving quadrants and nonunique representations.

    • Convert between Cartesian and polar coordinates.
    • Interpret negative radius and coterminal angles.
    • Handle the origin and quadrant ambiguity.

    6 practice tasks.

13. Trace polar curves instead of treating radius as height

    Lesson details and objectives

    Recognize circles, signed radii and repeated tracing through coordinated tables.

    • Trace a curve from polar samples and convert selected equations.
    • Distinguish radial change from Cartesian vertical change.
    • Identify repeated points and omitted cases in algebraic conversion.

    6 practice tasks.

14. Multiply complex numbers by scaling and rotating

    Lesson details and objectives

    Use modulus, argument, De Moivre and complete root families.

    • Convert complex numbers between rectangular and polar form.
    • Apply multiplication and integer-power formulas.
    • Find all nth roots and distinguish argument branches.

    6 practice tasks.

15. Separate a path from the way it is traveled

    Lesson details and objectives

    Use paired coordinate functions, parameter domains and tracing direction.

    • Eliminate parameters while preserving restrictions.
    • Distinguish geometric locus, direction and repeated traversal.
    • Represent lines, circles and ellipses parametrically.

    6 practice tasks.

16. Read conics from equations and geometric definitions

    Lesson details and objectives

    Complete squares and distinguish ellipses, hyperbolas and parabolas.

    • Identify centres, vertices and axis lengths from standard forms.
    • Connect parabolas to focus/directrix distance.
    • State limits of introductory unrotated-conic classification.

    6 practice tasks.