# 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

- Use the following compound angle identities:

[latex]\scriptsize \sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta[/latex]

[latex]\scriptsize \cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta[/latex]

to derive and apply the following double angle identities:

[latex]\scriptsize \sin 2\alpha =2\sin \alpha \cos \alpha[/latex]

[latex]\scriptsize \cos 2\alpha =\left\{ \begin{{\cos }^{2}}\alpha -{{\sin }^{2}}\alpha \\2{{\cos }^{2}}\alpha -1\\1-2{{\sin }^{2}}\alpha \end{align*} \right\}[/latex] - Determine the specific solutions of trigonometric expressions using compound and double angle identities without a calculator (e.g. [latex]\scriptsize \sin {{120}^\circ}[/latex], [latex]\scriptsize \cos {{75}^\circ}[/latex], etc.).
- Use compound angle identities to simplify trigonometric expressions and to prove trigonometric equations.
- Determine the specific solutions of trigonometric equations by using knowledge of compound angles and identities.

**Note:**- Solutions: [latex]\scriptsize \left[ {{{0}^\circ},{{{360}}^\circ}} \right][/latex].
- Identities limited to:

[latex]\scriptsize \tan \theta =\displaystyle \frac{{\sin \theta }}{{\cos \theta }}[/latex] and [latex]\scriptsize {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1[/latex]. - Double and compound angle identities are included.
- Radians are excluded.

- Solve problems from a given diagram in two and three dimensions by applying the sine and cosine rule.

**Note:**Area formula and compound angle identities are excluded.

### Unit 1 outcomes

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

- Expand the compound angles of [latex]\scriptsize \sin (\alpha \pm \beta )[/latex] and [latex]\scriptsize \cos (\alpha \pm \beta )[/latex].
- Use the compound expansions to simplify expressions.
- Use compound angles to prove identities.

### Unit 2 outcomes

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

- Solve equations involving double and compound angles.

### Unit 3 outcomes

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

- Apply the sine rule correctly to solve 2-D and 3-D problems.
- Apply the cosine rule correctly to solve 2-D and 3-D problems.

### Unit 4 outcomes

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

- Define radian measure.
- Convert from degrees to radians.
- Convert from radians to degrees.