Given r = 10 - 10 sin theta find the maximum value of |r| and zeros of r

Question:

Given {eq}r = 10 - 10 \sin \theta {/eq} find the maximum value of {eq}|r| {/eq} and zeros of {eq}r {/eq}

{eq}\sin {/eq} function

  • {eq}\sin \theta {/eq} is a continuous function defined for all values of {eq}\theta {/eq} .
  • The range of values of {eq}\sin \theta {/eq} is {eq}[-1,1] {/eq} .
  • The maximum value of {eq}\sin \theta {/eq} is {eq}1 {/eq} and its minimum value is {eq}-1 {/eq} .

Answer and Explanation:

  • Given {eq}r = 10 - 10 \sin \theta {/eq}

We know that {eq}\sin \theta \in [-1,1] {/eq}.

{eq}\begin{align} \text{Now } r &= 10 - 10 \sin \theta \\ &=10-10[-1,1]\\ &=10-[-10,10]\\ &=[0,20]\\ \therefore r\in [0,20] . \end{align} {/eq}

Hence the maximum value of {eq}r {/eq} is {eq}20 {/eq}.

  • For zeros of {eq}r {/eq} ; {eq}r =0 {/eq}

{eq}r = 10 - 10 \sin \theta =0 \\ \implies \sin \theta =1. {/eq}

Therefore the general solution of {eq}\sin \theta =1 {/eq} is {eq}\displaystyle \theta = n\pi+(-1)^n\frac{\pi}{2} {/eq} .


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Find the Maximum Value of a Function: Practice & Overview

from CAHSEE Math Exam: Tutoring Solution

Chapter 10 / Lesson 12
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