Trigonometry - Trigonometric Graphs and Parameters

Let \(f(x) = 3 \sin{x} + 4\cos{x}\), for \(-2\pi \leq x \leq 2\pi\).

1. Sketch the graph of \(f\).[3]
2. Write down
   1. the amplitude;
   2. the period;
   3. the \(x\)-intercept that lies between \(-\frac{\pi}{2}\) and \(0\).[3]
3. Hence write \(f(x)\) in the form \(p \sin{(qx + r)}\).[3]
4. Write down one value of \(x\) such that \(f'(x) = 0\).[2]
5. Write down the two values of \(k\) for which the equation \(f(x) = k\) has exactly two solutions.[2]
6. Let \(g(x) = \ln{(x +1)}\), for \(0 \leq x \leq \pi\). There is a value of \(x\), between \(0\) and \(1\), for which the gradient of \(f\) is equal to the gradient of \(g\). Find this value of \(x\).[5]