Trigonometric Identities Question 326
Question: Which of the following number(s) is/are rational
[IIT 1998]
Options:
A) $ \sin 15{}^\circ $
B) $ \cos 15{}^\circ $
C) $ \sin 15{}^\circ \cos 15{}^\circ $
D) $ \sin 15{}^\circ \cos 75{}^\circ $
Show Answer
Answer:
Correct Answer: C
Solution:
$ \sin 15^{o}=\sin (45^{o}-30^{o})=\frac{\sqrt{3}-1}{2\sqrt{2}}= $ irrational $ \cos 15^{o}=\cos (45^{o}-30^{o})=\frac{\sqrt{3}+1}{2\sqrt{2}} $ =irrational
$ \therefore ,\sin 15^{o}\cos 15^{o}=\frac{1}{2}(2\sin 15^{o}\cos 15^{o}) $ $ =\frac{1}{2}\sin 30^{o}=\frac{1}{2}.\frac{1}{2}=\frac{1}{4} $ = rational \ $ \sin 15^{o}\cos 75^{o}=\sin 15^{o}\sin 15^{o}={{\sin }^{2}}15^{o} $ $ ={{( \frac{\sqrt{3}-1}{2\sqrt{2}} )}^{2}}=\frac{4-2\sqrt{3}}{8} $ = irrational
BETA
