UPSEE Solved Model Paper 6

Here we can find which one is rational from given options. We solve each options from using suitable trigonometrical identities and find which one is rational. sin 15° = sin (45 - 30) = sin 45 cos 30 - cos 45 sin 30 = $\frac{1}{\sqrt{2}} \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \frac{1}{2}$ = $rac{\sqrt{3}-1}{2\sqrt{2}}$ Therefore, it is irrational. (Since, Sin(A - B) = sinAcosB - cosAsinB) Cos 150= Cos (45 – 30) = Cos 45 Cos 30 + Sin 45 Sin 30 = $\frac{1}{\sqrt{2}} \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \frac{1}{2}$ = $rac{\sqrt{3}+1}{2\sqrt{2}}$ Therefore, it is irrational sin 15° cos 15° = $rac{1}{2}(2 \sin 15^{\circ} \cos 15^{\circ})$ = $rac{1}{2} \sin 30$ = $\frac{1}{2} \times \frac{1}{2}$ = $rac{1}{4}$ Therefore, it is rational sin 15° cos 75° = sin 15° cos (90 - 15) = sin 5° . sin 15° = $\sin^2 15$ = $\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)^2$ Therefore , it is irrational sin 2A = 2 sin A cos A cos (90 - 15) = sin 15° Hence sin 15° . cos 15° is a rational