The function \(f\) is defined by \(f(x) = 3 \tan\left(\frac{1}{2}x\right) - 2\), for \(-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi\).
(i) Solve the equation \(f(x) + 4 = 0\), giving your answer correct to 1 decimal place. [3]
(ii) Find an expression for \(f^{-1}(x)\) and find the domain of \(f^{-1}\). [5]
(iii) Sketch, on the same diagram, the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\). [3]
A function \(f\) is defined by \(f : x \mapsto 5 - 2 \sin 2x\) for \(0 \leq x \leq \pi\).
(i) Find the range of \(f\). [2]
(ii) Sketch the graph of \(y = f(x)\). [2]
(iii) Solve the equation \(f(x) = 6\), giving answers in terms of \(\pi\). [3]
The function \(g\) is defined by \(g : x \mapsto 5 - 2 \sin 2x\) for \(0 \leq x \leq k\), where \(k\) is a constant.
(iv) State the largest value of \(k\) for which \(g\) has an inverse. [1]
(v) For this value of \(k\), find an expression for \(g^{-1}(x)\). [3]
The function f is defined by \(f : x \mapsto 4 \sin x - 1\) for \(-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi\).
The function \(f : x \mapsto 5 + 3 \cos\left(\frac{1}{2}x\right)\) is defined for \(0 \leq x \leq 2\pi\).
The function \(f : x \mapsto 6 - 4\cos\left(\frac{1}{2}x\right)\) is defined for \(0 \leq x \leq 2\pi\).
A function f is defined by \(f : x \mapsto 3 \cos x - 2\) for \(0 \leq x \leq 2\pi\).
A function g is defined by \(g : x \mapsto 3 \cos x - 2\) for \(0 \leq x \leq k\).
Prove the identity \(\frac{\tan^2 \theta + 1}{\tan^2 \theta - 1} \equiv \frac{1}{1 - 2 \cos^2 \theta}\).
Prove the identity \(\frac{1 - 2 \sin^2 \theta}{1 - \sin^2 \theta} \equiv 1 - \tan^2 \theta\).
Prove the identity \(\left( \frac{1}{\cos x} - \tan x \right) \left( \frac{1}{\sin x} + 1 \right) \equiv \frac{1}{\tan x}\).
Show that \(\frac{\sin \theta}{1 - \sin \theta} - \frac{\sin \theta}{1 + \sin \theta} \equiv 2 \tan^2 \theta\).
Show that \(\frac{\tan \theta}{1 + \cos \theta} + \frac{\tan \theta}{1 - \cos \theta} \equiv \frac{2}{\sin \theta \cos \theta}\).
Prove the identity \(\frac{1 + \sin \theta}{\cos \theta} + \frac{\cos \theta}{1 + \sin \theta} \equiv \frac{2}{\cos \theta}\).
Prove the identity \(\left( \frac{1}{\cos x} - \tan x \right)^2 \equiv \frac{1 - \sin x}{1 + \sin x}\).
Show that \(\frac{\tan \theta + 1}{1 + \cos \theta} + \frac{\tan \theta - 1}{1 - \cos \theta} \equiv \frac{2(\tan \theta - \cos \theta)}{\sin^2 \theta}\).
Prove the identity \((\sin \theta + \cos \theta)(1 - \sin \theta \cos \theta) \equiv \sin^3 \theta + \cos^3 \theta\).
Prove the identity \(\left( \frac{1}{\cos \theta} - \tan \theta \right)^2 \equiv \frac{1 - \sin \theta}{1 + \sin \theta}\).
Prove the identity \(\frac{1 + \cos \theta}{\sin \theta} + \frac{\sin \theta}{1 + \cos \theta} \equiv \frac{2}{\sin \theta}\).
Prove the identity \(\frac{\sin \theta}{\cos \theta + \sin \theta} + \frac{1 - \cos \theta}{\cos \theta - \sin \theta} \equiv \frac{\cos \theta + \sin \theta - 1}{1 - 2 \sin^2 \theta}\).
Show that \(\cos^4 x \equiv 1 - 2 \sin^2 x + \sin^4 x\).
Prove the identity \(\frac{1 + \cos \theta}{1 - \cos \theta} - \frac{1 - \cos \theta}{1 + \cos \theta} \equiv \frac{4}{\sin \theta \tan \theta}\).