# 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:
$\scriptsize \sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$
$\scriptsize \cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta$
to derive and apply the following double angle identities:
$\scriptsize \sin 2\alpha =2\sin \alpha \cos \alpha$
\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\}
•  Determine the specific solutions of trigonometric expressions using compound and double angle identities without a calculator (e.g. $\scriptsize \sin {{120}^\circ}$, $\scriptsize \cos {{75}^\circ}$, 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: $\scriptsize \left[ {{{0}^\circ},{{{360}}^\circ}} \right]$.
• Identities limited to:
$\scriptsize \tan \theta =\displaystyle \frac{{\sin \theta }}{{\cos \theta }}$ and $\scriptsize {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$.
• 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 $\scriptsize \sin (\alpha \pm \beta )$ and $\scriptsize \cos (\alpha \pm \beta )$.
• 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.