Exercise 9.1 — Question 4

Domain and Range of Trigonometric Functions

This question focuses on identifying the domain and range of the six trigonometric functions, and classifying them as even, odd, or neither, along with their periods.

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Domain and Range Summary

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Even and Odd Trigonometric Functions

A function $f$ is even if $f(-x)=f(x)$ for all $x$ in its domain (symmetric about the $y$-axis).

A function $f$ is odd if $f(-x)=-f(x)$ for all $x$ in its domain (symmetric about the origin).

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Periodicity of Trigonometric Functions

A function $f$ is periodic with period $T$ if $f(x+T)=f(x)$ for all $x$, and $T$ is the smallest such positive number.

Key reasoning:

• $\sin (x+2\pi )=\sin x$ and $\cos (x+2\pi )=\cos x$, so their period is $2\pi$.
• $\tan (x+\pi )=\tan x$, so the period of $\tan x$ is $\pi$.

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Worked Example

Q: State whether $f(x)=\sin x\cdot \cos x$ is even, odd, or neither. Also find its period.

Solution:

Even/Odd check: $f(-x)=\sin (-x)\cdot \cos (-x)=(-\sin x)(\cos x)=-\sin x\cos x=-f(x)$

Since $f(-x)=-f(x)$, the function is odd.

Period: $f(x)=\sin x\cos x=\frac{1}{2}\sin 2x$

The period of $\sin 2x$ is $\frac{2\pi }{2}=\pi$, so the period of $f(x)$ is $\pi$.