First principles: trigonometric functions

7 Differentiate \(\cos x\) with respect to \(x\), from first principles.

9. Given that \(x\) is measured in radians, prove, from the first principles, that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sin x ) = \cos x$$ You may assume the formula for \(\sin ( A \pm B )\) and that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\).

12. $$y = \sin x$$ where \(x\) is measured in radians.
Use differentiation from first principles to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cos x$$ You may

• use without proof the formula for \(\sin ( A \pm B )\)
• assume that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\)

1. Given that \(\theta\) is measured in radians, prove, from first principles, that

$$\frac { \mathrm { d } } { \mathrm {~d} \theta } ( \cos \theta ) = - \sin \theta$$ You may assume the formula for \(\cos ( A \pm B )\) and that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\) (5)

\(f(x) = \sin x\) Using differentiation from first principles find the exact value of \(f'\left(\frac{\pi}{6}\right)\) Fully justify your answer. [6 marks]