Question

Calculate $\sin 18$ and $\cos 18$ without using trigonometric table.

Answer 1

Let $5A=90$

$2A+3A=90$

$2A=90-3A$

Apply $\sin$ on both sides, we get

$\sin \left(2A\right)=\sin (90-3A)$

$\Rightarrow \sin \left(2A\right)=\cos \left(3A\right)$

$\Rightarrow \sin \left(2A\right)=4{\cos }^{3}A-3\cos \left(A\right)$

$\Rightarrow 2\sin \left(A\right)\cos \left(A\right)=4{\cos }^3\left(A\right)-3\cos \left(A\right)$

$\Rightarrow \cos \left(A\right)\left[2\sin \right(A)-4{\cos }^{2}A+3]=0$

As $A=18,\cos \left(A\right)\ne 0$

$\therefore 2\sin \left(A\right)-4(1-{\sin }^2(A\left)\right)+3=0$

$\Rightarrow 4{\sin }^{2}A+2\sin \left(A\right)-1=0$

$\therefore \sin \left(A\right)=\frac{-2\pm \sqrt{4-4\cdot 4(-1)}}{2.4}$

$\Rightarrow \sin \left(18\right)=\frac{-1\pm \sqrt{5}}{4}$

As $18$ lies in 1st quadrant and $\sin$ is positive in 1st quadrant ;

$\therefore \sin \left(18\right)=\frac{-1+\sqrt{5}}{4}$

$\cos \left(18\right)=1-{\sin }^{2}18$

$\Rightarrow \cos \left(18\right)=\frac{\sqrt{10+2\sqrt{5}}}{4}$