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.

The answer is (d).