# 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 $\scriptsize {{30}^\circ}$, $\scriptsize {{45}^\circ}$ and $\scriptsize {{60}^\circ}$.
• 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:
• $\scriptsize \sin ({{90}^\circ}\pm \theta )$; $\scriptsize \cos ({{90}^\circ}\pm \theta )$
• $\scriptsize \sin ({{180}^\circ}\pm \theta )$; $\scriptsize \cos ({{180}^\circ}\pm \theta )$; $\scriptsize \tan ({{180}^\circ}\pm \theta )$
• $\scriptsize \sin ({{360}^\circ}\pm \theta )$; $\scriptsize \cos ({{360}^\circ}\pm \theta )$; $\scriptsize \tan ({{360}^\circ}\pm \theta )$
• Use the following trigonometric identities to simplify expressions and prove equations:
• $\scriptsize \tan \theta =\displaystyle \frac{{\sin \theta }}{{\cos \theta }}$
• $\scriptsize {{\sin }^{2}}\theta +co{{s}^{2}}\theta =1$
• Solve trigonometric equations (with the use of a calculator) involving reduction formulae using special triangles for the three trigonometric functions in all four quadrants.
• $\scriptsize \sin ({{90}^\circ}\pm \theta )$; $\scriptsize \cos ({{90}^\circ}\pm \theta )$
• $\scriptsize \sin ({{180}^\circ}\pm \theta )$; $\scriptsize \cos ({{180}^\circ}\pm \theta )$; $\scriptsize \tan ({{180}^\circ}\pm \theta )$
• $\scriptsize \sin ({{360}^\circ}\pm \theta )$; $\scriptsize \cos ({{360}^\circ}\pm \theta )$; $\scriptsize \tan ({{360}^\circ}\pm \theta )$
• 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 $\scriptsize {{30}^\circ}$, $\scriptsize {{45}^\circ}$ and $\scriptsize {{60}^\circ}$.

### Unit 2 outcomes

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

• Apply reduction formulae for function values of $\scriptsize ({{90}^\circ}\pm \theta )$.
• Apply reduction formulae for function values of $\scriptsize ({{180}^\circ}\pm \theta )$.
• Apply reduction formulae for function values of $\scriptsize ({{360}^\circ}\pm \theta )$.
• 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 $\scriptsize \tan \theta =\displaystyle \frac{{\sin \theta }}{{\cos \theta }}$ to prove equations and simplify expressions.
• Use the square identity $\scriptsize {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$ 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.