Find the area of triangle AOP in terms of r and $\theta$ .

Find the area of triangle POT in terms of r and $\theta$ .

Using your results from part (a) and part (b), show that $\sin \theta < \theta < \tan \theta$ .

area of ${\text{AOP}} = \frac{1}{2}{r^2}\sin \theta$     A1

[1 mark]

${\text{TP}} = r\tan \theta$     (M1)

area of POT $= \frac{1}{2}r(r\tan \theta )$

$= \frac{1}{2}{r^2}\tan \theta$     A1

[2 marks]

area of sector OAP $= \frac{1}{2}{r^2}\theta$     A1

area of triangle OAP < area of sector OAP < area of triangle POT     R1

$\frac{1}{2}{r^2}\sin \theta < \frac{1}{2}{r^2}\theta < \frac{1}{2}{r^2}\tan \theta$

$\sin \theta < \theta < \tan \theta$     AG

[2 marks]

The majority of candidates were able to find the area of Triangle AOP correctly. Most were then able to get an expression for the other triangle. In the final section, few saw the connection between the area of the sector and the relationship. 

The majority of candidates were able to find the area of Triangle AOP correctly. Most were then able to get an expression for the other triangle. In the final section, few saw the connection between the area of the sector and the relationship.

The majority of candidates were able to find the area of Triangle AOP correctly. Most were then able to get an expression for the other triangle. In the final section, few saw the connection between the area of the sector and the relationship.