Which of the following is greatest?
tan 1
All angles are in radians. Since \(\tan\) has period \(\pi\), reduce each angle to an equivalent angle in \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\).
\(\tan 1\): since \(1<\dfrac{\pi}{2}\approx1.5708\), this is already in the first quadrant close to \(\pi/2\), giving \(\tan 1\approx1.557\) (a relatively large positive value).
\(\tan 4=\tan(4-\pi)=\tan(0.858)\) (since \(\pi\approx3.1416\)), and \(\tan(0.858)\approx1.158\).
\(\tan 7=\tan(7-2\pi)=\tan(0.717)\) (since \(2\pi\approx6.2832\)), giving \(\tan(0.717)\approx0.871\).
\(\tan 10=\tan(10-3\pi)=\tan(0.575)\) (since \(3\pi\approx9.4248\)), giving \(\tan(0.575)\approx0.648\).
Comparing: \(\tan1\approx1.557>\tan4\approx1.158>\tan7\approx0.871>\tan10\approx0.648\).
Hence \(\tan 1\) is the greatest.
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