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

### Subject outcome

Subject outcome 3.3: Solve problems by constructing and interpreting trigonometric models

### Learning outcomes

- 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.

### Unit 1 outcomes

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].

### Unit 2 outcomes

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.

### Unit 3 outcomes

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.

### Unit 4 outcomes

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

- Solve trigonometric equations using a general solution.

### Unit 5 outcomes

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

- Apply the area rule in 2-D triangles.

### Unit 6 outcomes

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.

### Unit 7 outcomes

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.

### Unit 8 outcomes

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

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