$Which of the following is the correct function for the graph below?$

$if \tan \frac{\theta }{2}=\frac{1}{2}, find the value of \cos \theta in exact form$

$Evaluate \sin \left[2{\sin }^{-1}\frac{3}{4}\right]$

$$\begin{gathered} The equation of the curve above is: \\ \left(A\right) \mathcal{y}=2{\sin }^{-1}\frac{\mathcal{x}}{3} \\ \left(B\right) \mathcal{y}=3{\sin }^{-1}\frac{\mathcal{x}}{2} \\ \left(C\right) \mathcal{y}=2{\sin }^{-1}3\mathcal{x} \\ \left(D\right) \mathcal{y}=3{\sin }^{-1}3\mathcal{x} \end{gathered}$$

$$\begin{gathered} a) Conside the equation {\tan }^{-1}x-\frac{x}{2}=0 \\ U\sin g x=2 as the first approximation to one of the roots, \\ use one application of Newton\text{'}s method to find a better \\ approximation to his root, correct to 2 decimal places. \end{gathered}$$

$$\begin{gathered} b) Find the exact volume of the solid obtained by rotating \\ y={\cos }^{-1}x about the y-axis between y=0 and y=\mathrm{\pi } \end{gathered}$$

$$\begin{gathered} Consider the function f\left(x\right) =2 {\sin }^{-1}\sqrt{x}-{\sin }^{-1}(2x-1) \\ i) Find the domain of f(x). \\ ii) show that f\text{'}\left(x\right)=0 \\ iii) Show that the graph of y=f(x). \end{gathered}$$

$$\begin{gathered} Prove \sin (A+B)+\sin (A-B)=2\sin A\cos B \\ Hence or otherwise show that {\int }_0^t \sin \left(\theta \mathcal{x}\right)\cos \left(\theta \right(t-\mathcal{x}\left)\right)d\mathcal{x}=\frac{1}{2}t\sin \left(\theta t\right) \end{gathered}$$

$Sketch \mathcal{Y}=\left({\cos }^{-1}\mathcal{x}\right)-\frac{\mathrm{\pi }}{2} showing all impor\tan t features$

$$\begin{gathered} Differentiate with respect to x \\ \left(i\right) {\cos }^{-1} 3x \\ \left(ii\right) {\tan }^2 3x \end{gathered}$$