Exercise 9.1 — Question 6

Question

Find the domain and range of the following trigonometric functions, state whether each is even or odd, give its period, and sketch its graph:

$y=\sin \theta ,y=\cos \theta ,y=\tan \theta$

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Domain and Range

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Even and Odd Functions

A function $f$ is even if $f(-x)=f(x)$ and odd if $f(-x)=-f(x)$.

• $\cos (-\theta )=\cos \theta$ → $y=\cos \theta$ is an even function.
• $\sin (-\theta )=-\sin \theta$ → $y=\sin \theta$ is an odd function.
• $\tan (-\theta )=-\tan \theta$ → $y=\tan \theta$ is an odd function.

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Periodicity

A function $f$ is periodic with period $T$ if $f(\theta +T)=f(\theta )$ for all $\theta$.

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Key Graph Properties

$y=\sin \theta$

• Amplitude: 1 (maximum value = 1, minimum value = −1)
• Period: $2\pi$
• Zeros: $\theta =n\pi ,n\in \mathbb{Z}$
• Odd function — graph is symmetric about the origin
• Passes through $(0,0)$, reaches maximum at $\theta =\frac{\pi }{2}$, minimum at $\theta =\frac{3\pi }{2}$

$y=\cos \theta$

• Amplitude: 1 (maximum value = 1, minimum value = −1)
• Period: $2\pi$
• Zeros: $\theta =(2n+1)\frac{\pi }{2},n\in \mathbb{Z}$
• Even function — graph is symmetric about the $y$-axis
• Passes through $(0,1)$, reaches minimum at $\theta =\pi$

$y=\tan \theta$

• No amplitude (range is all real numbers)
• Period: $\pi$
• Zeros: $\theta =n\pi ,n\in \mathbb{Z}$
• Odd function — graph is symmetric about the origin
• Has vertical asymptotes at $\theta =(2n+1)\frac{\pi }{2}$
• Increasing on each interval between consecutive asymptotes

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Summary Table