Which of these log and trig expressions is the largest?

Which of these log and trig expressions is the largest?

Question

Which of the following numbers is largest in value? (All angles are given in radians.)

1. $\tan\left(\frac{5\pi}{4}\right)$,

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1. $\sin^2 \left(\frac{5\pi}{4}\right)$,

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1. $\log_{10} \left(\frac{5\pi}{4}\right)$,

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1. $\log_2 \left(\frac{5\pi}{4}\right)$.

Suggestion

Which of the terms can we evaluate exactly? How do the other terms compare?

Solution

Let’s consider the numbers in order.

a: Using the fact that $\tan$ has period $\pi$, we have $\tan \left(\frac{5\pi}{4}\right)=\tan\left(\pi + \frac{\pi}{4}\right) = \tan \left(\frac{\pi}{4}\right)=1 .$

b: Since

$\sin(\pi - x) = \sin(x)$

, we have

$$\begin{align*} \sin\left(\frac{5\pi}{4}\right)&=\sin\left(\pi-\frac{5\pi}{4}\right)\\ &=\sin\left(\frac{-\pi}{4}\right)\\ &=-\sin\left(\frac{\pi}{4}\right) = \frac{-1}{\sqrt{2}}\\ \implies \sin^2 \left(\frac{5\pi}{4}\right)&=\frac{1}{2} . \end{align*}$$

c: Now, since $\pi<4$, we know that $\frac{5\pi}{4}<5$ and so clearly $\frac{5\pi}{4}<10$. Therefore, $\log_{10}\left(\frac{5\pi}{4}\right)<1 .$

d: Similarly, $\pi>3 \implies \frac{5\pi}{4}>\frac{15}{4}>2$. Hence, $\log_{2}\left(\frac{5\pi}{4}\right)>1.$ So this expression has the largest value of the four.

The answer is (d).