Define the function by

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Define the function

f:\,\text{Dom}(f)\to\mathbb R

f(x)=\sin x + \tan x

, where

\text{Dom}(f)

is the largest subset of

\mathbb R

on which

f

is defined. (i) Determine

\text{Dom}(f)

and write it as a union of open intervals. (ii) Prove that

f(x)

is unbounded above and unbounded below on every interval forming its domain. (iii) Show that in the interval

\left(0,\dfrac{\pi}{2}\right)

the equation

f(x)=2

has exactly one real root. Approximate this root correct to two decimal places.