Paper 2

(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.

(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.