Given $\sin(x)$, find $\sin(\frac{x}{2}) \cos(\frac{5x}{2})$

Question

Knowing that $\pi < 2x < 2\pi$ and

$$\sin(x) = \frac{4}{5},$$

find

$$\sin\left(\frac{x}{2}\right) \cos\left(\frac{5x}{2}\right)\ =\ ?$$

Answer 1

HINT

Use that

$$\sin \theta \cos \varphi = \frac12{{\sin(\theta + \varphi) + \frac12 \sin(\theta - \varphi)} }$$

and

• $\sin (2\theta) =2\sin\theta \cos\theta$

• $\sin (3\theta) =3\sin\theta - 4\sin^3\theta$

moreover from the given

• $\cos x=-\sqrt{1-\sin^2 x}$

Answer 2

$sin(x/2)cos(5x/2)\\ =sin(x/2)cos(2x+x/2)\\=sin(x/2)\{cos(2x)cos(x/2)-sin(2x)sin(x/2)\}\\=cos(2x)cos(x/2)sin(x/2)-sin(2x)sin^2(x/2)\\=1/2(1-2sin^2(x))sin(x)-sin(x)cos(x)(1-cos(x))$

As $\pi/2<x<\pi$, $sin(x)=4/5$ implies $cos(x)=-3/5$. So the final answer is $sin(x/2)cos(5x/2)=82/125.$