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Paper 2

TRIGONOMETRY
(a) Derive and use the values of the trigonometric functions (in surd form where applicable) of 30º, 45ºand 60º.

(b) Derive and use the following identities:

and
sin2θ + cos2θ = 1

(c) Derive the reduction formulae for:
      sin(90° ± θ)
      cos(90° ± θ)

      sin(180° ± θ)
      cos(180° ± θ)
      tan(180° ± θ)

      sin(360° ± θ)
      cos(360° ± θ)
      tan(360° ± θ)

      sin(-θ)
      cos(-θ)
      tan(-θ)

(d) Determine the general solution of trigonometric equations

(e) Establish and apply the sine, cosine and area rules.

Solve problems in two dimensions by using the sine, cosine and area rules; and by constructing and interpreting geometric and trigonometric models.

Generate as many graphs as necessary, initially by means of point-by-point plotting, supported by available technology, to make and test conjectures about the effect of the parameters k, p, a and q for functions including:
      y = sinkx
      y = coskx
      y = tankx
      y = sin(x + p)
      y = cos(x + p)
      y = tan(x + p)

Identify characteristics as listed below and hence use applicable characteristics to sketch graphs of functions including those listed above:
(a) domain and range;
(b) intercepts with the axes;
(c) turning points, minima and maxima;
(d) asymptotes;
(e) shape and symmetry;
(f) periodicity and amplitude;
(g) average gradient (average rate of change);
(h) intervals on which the function increases/decreases;
(i) the discrete or continuous nature of the graph.

Clarification

  • Simplify and solve Pythagorean trigonometric problems using the definitions of trigonometric functions.
  • Simplify expressions and prove trigonometric identities involving
    - Reduction formulae
    - Special angles
    - Negative angles
    - Complementary ratios (sin25°=cos65°) and using the identities

    and sin2θ + cos2θ = 1

  • Solve trigonometric equations with or without the use of a calculator and determining both general and specific solutions to the equation. Determining the solution to a trigonometric equation can be integrated with a graph question, specifically determining the point of intersection or in the form of an inequality.
  • Solution of non-right-angled triangles specifically including
    - Area formula
    - Sine rule
    - Cosine rule
    - Solve 2-D problems using the above rules.
    NOTE:
    Proofs are NOT required for examination purposes but should be part of the learning process to enhance understanding.
  • The focus of trigonometric graphs in paper 2 is on the relationships, simplification and determining points of intersection by solving equations, although the characteristics of the graphs should not be excluded.


Derive and use the following compound angle identities:
(a)

(b)

(c)

(d)

Solve problems in two and three dimensions by constructing and interpreting geometric and trigonometric models. Clarification

  • Use the compound angle formula for:
    cos(α – β)
    and derive the formulae for:
    sin(α ± β) and
    cos(α + β).
    Note:
    Proofs are NOT required for examination purposes but should be part of the learning process to enhance understanding.
  • Use the compound angle formulae in
    - Simplifying trigonometric expressions
    - Proving identities
    - Solving trigonometric equations (both specific and general solutions)
    - Solving trigonometric equations where the denominator of an identity is undefined.
    - Integration with transformation geometry.
  • Solution of non-right-angled triangles specifically including
    - Area formula
    - Sine
    - Cosine rule
    - Solve 2-D & 3-D problems using the above rules.