Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 210: 101b


$I$ reaches its maximum value as $\cos^2\theta$ reaches its maximum value, which is $1$. At $\theta=0$, $\cos^2\theta=1$, so at $\theta=0$, $I$ reaches its maximum value.

Work Step by Step

$$I=k\cos^2\theta$$ From the formula of Lambert's Law, we see that $I$ would reach its maximum value when $k\cos^2\theta$ also reaches its maximum value. However, since $k$ is a constant and as a result, does not change its value, the maximum value of $k\cos^2\theta$ happens at the maximum value of $\cos^2\theta$. Overall, the maximum value of $I$ occurs when $\cos^2\theta$ reaches its maximum value. We remember that the range of $\cos\theta$ is $[-1,1]$. In other words, $$-1\le\cos\theta\le1$$ That means, $$0\le\cos^2\theta\le1$$ Therefore, the maximum value of $\cos^2\theta$ is $1$. We find that as $\theta=0$, $\cos^2\theta=1$, which is the maximum value of $\cos^2\theta$, meaning also the maximum value of $I$ has been reached. Therefore, the maximum value of $I$ occurs when $\theta=0$.
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