Review question

# Which of these log and trig expressions is the largest? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7184

## Solution

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

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

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

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

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

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.