9.1 Q-1

Exercise 9.1 — Domain and Range of Trigonometric Functions

This exercise covers finding the domain and range of all six trigonometric functions.

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Summary Table: Domain and Range

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Key Reasoning

Why is the domain of $\tan x$ restricted? $\tan x=\frac{\sin x}{\cos x}$ This is undefined when $\cos x=0$, i.e., at $x=\frac{(2n+1)\pi }{2}$.

Why is the domain of $\cot x$ restricted? $\cot x=\frac{\cos x}{\sin x}$ This is undefined when $\sin x=0$, i.e., at $x=n\pi$.

Why is the range of $\sec x$ and $\csc x$ restricted?

• Since $|\cos x|\le 1$, we have $|\sec x|=\frac{1}{|\cos x|}\ge 1$.
• Similarly $|\csc x|\ge 1$.
• Therefore both functions take values in $(-\infty ,-1]\cup [1,+\infty )$.

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Worked Example

Q: Find the domain and range of $y=\sin x$.

Solution:

• $\sin x$ is defined for every real number $x$, so Domain $=\mathbb{R}$.
• The sine function oscillates between $-1$ and $1$ inclusive, so Range $=[-1,1]$.

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