Mistakes give rise to Problems- 11

We have the formula $\sin(3a) = 3\sin (a) - 4\sin^3 (a)$

Hence asked property is true if and only if $4\sin^3 (a)=0$ , i.e. $\sin(a)=0$ .

This is true is $a= n\pi$ where $n\in \mathbb{Z}$

As we know

$\displaystyle \lfloor \dfrac{100}{\pi} \rfloor = 31$ ,

There are 31 multiples if $\pi$ in $(0,100]$ and hence $31$ multiples in $[-100,0)$ and the cases when $a=0\times \pi$ ,

so there are $\boxed{63}$ such angles.

Let x = sin 3x , 3x = 3x - 4x^3. x = 0, therefore sinx = 0 one value of x = 0. The solution set of this becomes n*pi where n is an integer. if n =32, the value will be greater than 100. therefore, the values of n will be in the interval [-31,31] which will yield to 2(31) + 1 = 63 angles.