Space, shape and measurement: Solve problems by constructing and interpreting trigonometric models

• Derive and use the values of the trigonometric functions (in surd form where applicable) of [latex]\scriptsize {{30}^\circ}[/latex], [latex]\scriptsize {{45}^\circ}[/latex] and [latex]\scriptsize {{60}^\circ}[/latex].
• Use the reduction formulae and special angles to solve trigonometric expressions and prove equations in all four quadrants (without the use of a calculator) for the following functions:
  • [latex]\scriptsize \sin ({{90}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \cos ({{90}^\circ}\pm \theta )[/latex]
  • [latex]\scriptsize \sin ({{180}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \cos ({{180}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \tan ({{180}^\circ}\pm \theta )[/latex]
  • [latex]\scriptsize \sin ({{360}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \cos ({{360}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \tan ({{360}^\circ}\pm \theta )[/latex]
• Use the following trigonometric identities to simplify expressions and prove equations:
  • [latex]\scriptsize \tan \theta =\displaystyle \frac{{\sin \theta }}{{\cos \theta }}[/latex]
  • [latex]\scriptsize {{\sin }^{2}}\theta +co{{s}^{2}}\theta =1[/latex]
• Solve trigonometric equations (with the use of a calculator) involving reduction formulae using special triangles for the three trigonometric functions in all four quadrants.
  • [latex]\scriptsize \sin ({{90}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \cos ({{90}^\circ}\pm \theta )[/latex]
  • [latex]\scriptsize \sin ({{180}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \cos ({{180}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \tan ({{180}^\circ}\pm \theta )[/latex]
  • [latex]\scriptsize \sin ({{360}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \cos ({{360}^\circ}\pm \theta )[/latex]; [latex]\scriptsize \tan ({{360}^\circ}\pm \theta )[/latex]
• Apply the sine, cosine and area rules.
• Solve problems in two dimensions by using the sine, cosine and area rules by interpreting given geometric and trigonometric models.

By the end of this unit you will be able to:

• Simplify trigonometric ratios involving the angles [latex]\scriptsize {{30}^\circ}[/latex], [latex]\scriptsize {{45}^\circ}[/latex] and [latex]\scriptsize {{60}^\circ}[/latex].

By the end of this unit you will be able to:

• Apply reduction formulae for function values of [latex]\scriptsize ({{90}^\circ}\pm \theta )[/latex].
• Apply reduction formulae for function values of [latex]\scriptsize ({{180}^\circ}\pm \theta )[/latex].
• Apply reduction formulae for function values of [latex]\scriptsize ({{360}^\circ}\pm \theta )[/latex].
• Simplify trigonometric ratios involving any combination of reduction formulae.

By the end of this unit you will be able to:

• Use the quotient identity [latex]\scriptsize \tan \theta =\displaystyle \frac{{\sin \theta }}{{\cos \theta }}[/latex] to prove equations and simplify expressions.
• Use the square identity [latex]\scriptsize {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1[/latex] to prove equations and simplify expressions.

By the end of this unit you will be able to:

• Solve trigonometric equations using a general solution.

By the end of this unit you will be able to:

• Apply the area rule in 2-D triangles.

By the end of this unit you will be able to:

• Use the sine rule to find an unknown length of a triangle.
• Use the sine rule to find an unknown angle of a triangle.

By the end of this unit you will be able to:

• Use the cosine rule to find the length of an unknown side of a triangle.
• Use the cosine rule to find an unknown angle of a triangle.

By the end of this unit you will be able to:

• Solve real problems by applying the area, sine and cosine rules.