Question 14(c) | BetaMath

Question 14(c)

(i) Explain why the equation $\tan^{–1}{\left(3x\right)} + \tan^{–1}{\left(10x\right)} = \theta$ , where $-\pi < \theta < \pi$ , has exactly one solution.

(ii) Solve $\displaystyle \tan^{–1}{\left(3x\right)} + \tan^{–1}{\left(10x\right)} = \frac{3\pi}{4}$

Solution

Part (i):

The function $\tan^{–1}{x}$ as monotonically increasing and has a range of is $\displaystyle \left(-\frac{\pi}2, \frac{\pi}2 \right)$ .

This means that both $\tan^{–1}{(3x)}$ and $\tan^{–1}{(10x)}$ are also monotonically increasing and have the range $\displaystyle \left(-\frac{\pi}2, \frac{\pi}2 \right)$ .

...