Exercise 9.1 — Question 2

This exercise focuses on finding the maximum and minimum values of trigonometric functions, and understanding the domain, range, and properties of $\sin \theta$, $\cos \theta$, and $\tan \theta$.

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Key Concepts

Domain and Range of Trigonometric Functions

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Maximum and Minimum Values

For a function of the form $y=a+b\sin (cx+d)$ or $y=a+b\cos (cx+d)$:

$\text{Maximum value}=a+|b|$ $\text{Minimum value}=a-|b|$

This is because $\sin$ and $\cos$ oscillate between $-1$ and $+1$.

Example: Find the maximum and minimum values of $y=5-3\cos (2\theta )$.

Here $a=5$, $b=-3$, so $|b|=3$.

$\text{Maximum}=5+3=8(\text{when }\cos (2\theta )=-1)$ $\text{Minimum}=5-3=2(\text{when }\cos (2\theta )=1)$

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Maximum and Minimum of Reciprocal Trigonometric Expressions

For $y=\frac{1}{f(\theta )}$ where $f(\theta )>0$:

• Maximum of $y$ $=\frac{1}{\min f(\theta )}$ (smallest denominator gives largest value)
• Minimum of $y$ $=\frac{1}{\max f(\theta )}$ (largest denominator gives smallest value)

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Properties of Graphs of $\sin \theta$, $\cos \theta$, and $\tan \theta$

$y=\sin \theta$:

• Period: $2\pi$
• Amplitude: $1$
• Odd function: $\sin (-\theta )=-\sin \theta$
• Passes through the origin $(0,0)$

$y=\cos \theta$:

• Period: $2\pi$
• Amplitude: $1$
• Even function: $\cos (-\theta )=\cos \theta$
• Maximum at $\theta =0$: $\cos (0)=1$

$y=\tan \theta$:

• Period: $\pi$
• No amplitude (range is all reals)
• Odd function: $\tan (-\theta )=-\tan \theta$
• Vertical asymptotes at $\theta =\frac{(2n+1)\pi }{2}$

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Even and Odd Trigonometric Functions

• Even functions satisfy $f(-x)=f(x)$: $\cos \theta$ and $\sec \theta$ are even.
• Odd functions satisfy $f(-x)=-f(x)$: $\sin \theta$, $\tan \theta$, $\csc \theta$, and $\cot \theta$ are odd.

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Flashcards

MCQs