Question

Select the correct statements for the corresponding values of the given trigonometric ratios.

Answer 1

We have to find the values of:
$\cos \frac{\pi }{12};\cos \frac{5\pi }{12};\sin \frac{\pi }{12}\&\sin \frac{5\pi }{12}$

We can write $\frac{\pi }{12}$ as: $\frac{\pi }{12}=\frac{\pi }{3}-\frac{\pi }{4}$

$\Rightarrow \sin \frac{\pi }{12}=\sin (\frac{\pi }{3}-\frac{\pi }{4})$

Using the compound angle formula for $\sin ,$ we get:

$\sin (\frac{\pi }{3}-\frac{\pi }{4})=\sin \frac{\pi }{3}\cos \frac{\pi }{4}-\cos \frac{\pi }{3}\sin \frac{\pi }{4}$
$\Rightarrow \sin (\frac{\pi }{3}-\frac{\pi }{4})=\frac{\sqrt{3}}{2}\cdot \frac{1}{\sqrt{2}}-\frac{1}{2}\cdot \frac{1}{\sqrt{2}}$
$\Rightarrow \sin (\frac{\pi }{3}-\frac{\pi }{4})=\frac{\sqrt{3}-1}{2\sqrt{2}}$

Similarly,
$\cos \frac{\pi }{12}=\cos (\frac{\pi }{3}-\frac{\pi }{4})$
$\Rightarrow \cos (\frac{\pi }{3}-\frac{\pi }{4})=\cos \frac{\pi }{3}\cos \frac{\pi }{4}+\sin \frac{\pi }{3}\sin \frac{\pi }{4}$
$\Rightarrow \cos (\frac{\pi }{3}-\frac{\pi }{4})=\frac{1}{2}\cdot \frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}\cdot \frac{1}{\sqrt{2}}$
$\Rightarrow \cos (\frac{\pi }{3}-\frac{\pi }{4})=\frac{\sqrt{3}+1}{2\sqrt{2}}$

Also, $\frac{5\pi }{12}=\frac{\pi }{2}-\frac{\pi }{12}$

$\Rightarrow \frac{5\pi }{12}\&\frac{\pi }{12}$ are complimentary angles.

$\Rightarrow \sin \frac{5\pi }{12}=\cos \frac{\pi }{12}\&\cos \frac{5\pi }{12}=\sin \frac{\pi }{12}$

$\Rightarrow \sin \frac{5\pi }{12}=\cos \frac{\pi }{12}=\frac{\sqrt{3}+1}{2\sqrt{2}}\&\cos \frac{5\pi }{12}=\sin \frac{\pi }{12}\frac{\sqrt{3}+1}{2\sqrt{2}}=\frac{\sqrt{3}-1}{2\sqrt{2}}$