Use the diagram to explain why[br][math]\sin\left(\theta\right)\approx\theta[/math] and [math]\tan\left(\theta\right)\approx\theta[/math].[br][br]Where is a line of length [math]\cos\theta[/math]?[br]Allowing that [math]\text{chord}\theta\approx\theta[/math] use pythagoras to derive [math]\cos\theta\approx1-\frac{1}{2}\theta^2[/math].

Explain why an enlargement of the [math]\tan\theta[/math] line by scale factor [math]\cos^2\theta[/math] appears as shown.[br]Use this to show [math]\tan\theta\cos^2\theta<\theta<\tan\theta[/math].[br][br]What can you deduce about the value of [math]\frac{\theta}{\tan\theta}[/math] as [math]\theta\to0[/math]?[br][br]Explain why an enlargement of the [math]\sin\theta[/math]line by scale factor [math]\frac{1}{\cos^2\theta}[/math] appears as shown.[br]Use this to show [math]\sin\theta<\theta<\frac{\sin\theta}{\cos^2\theta}[/math].[br][br]What can you deduce about the value of [math]\frac{\theta}{\sin\theta}[/math] as [math]\theta\to0[/math]?[br]