The problem consists of several calculus questions, including finding roots of equations, ranges and domains of functions, verifying trigonometric identities, solving trigonometric equations, determining if functions are odd or even, and finding derivatives. I will solve question 3a(i). The question asks to find all solutions to the problem $\sin(2\theta) = \tan(\theta)$.

1. Problem Description

The problem consists of several calculus questions, including finding roots of equations, ranges and domains of functions, verifying trigonometric identities, solving trigonometric equations, determining if functions are odd or even, and finding derivatives. I will solve question 3a(i). The question asks to find all solutions to the problem $\sin(2\theta) = \tan(\theta)$.

2. Solution Steps

We are asked to find all solutions to the equation $\sin(2\theta) = \tan(\theta)$.

We can use the double-angle identity for sine:

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$.

Then, our equation becomes $2\sin(\theta)\cos(\theta) = \tan(\theta)$.

Since $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, we can rewrite the equation as $2\sin(\theta)\cos(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

Rearranging, we get $2\sin(\theta)\cos(\theta) - \frac{\sin(\theta)}{\cos(\theta)} = 0$.

Factoring out $\sin(\theta)$, we have $\sin(\theta)\left(2\cos(\theta) - \frac{1}{\cos(\theta)}\right) = 0$.

This implies that either $\sin(\theta) = 0$ or $2\cos(\theta) - \frac{1}{\cos(\theta)} = 0$.

If $\sin(\theta) = 0$, then $\theta = n\pi$, where $n$ is an integer.

If $2\cos(\theta) - \frac{1}{\cos(\theta)} = 0$, then $2\cos^2(\theta) - 1 = 0$, which gives $2\cos^2(\theta) = 1$, so $\cos^2(\theta) = \frac{1}{2}$.

Then $\cos(\theta) = \pm\frac{1}{\sqrt{2}} = \pm\frac{\sqrt{2}}{2}$.

Thus, $\theta = \frac{\pi}{4} + \frac{n\pi}{2}$, where $n$ is an integer.

Combining the solutions, we have $\theta = n\pi$ and $\theta = \frac{\pi}{4} + \frac{n\pi}{2}$, where $n$ is an integer.

3. Final Answer

The solutions are $\theta = n\pi$ and $\theta = \frac{\pi}{4} + \frac{n\pi}{2}$, where $n$ is an integer.