# Trigo-system

#### dhiab

Solve : $$\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}$$
$$\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4}$$

#### Sudharaka

Solve : $$\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}$$
$$\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4}$$
Dear dhiab,

Use the trignometric identities given below. Then you would be able to solve your problem.

$$\displaystyle cos(A+B)=cosAcosB-sinAsinB$$

$$\displaystyle cos(A-B)=cosAcosB+sinAsinB$$

dhiab

#### Failure

Solve : $$\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}$$
$$\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4}$$
It follows that:
$$\displaystyle \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)=\frac{\sqrt{3}}{4}-\frac{\sqrt{3}}{4}=0$$

Hence,

$$\displaystyle x+y=\frac{\pi}{2}+n\pi, \qquad n\in\mathbb{Z}$$

Now plug $$\displaystyle y=\frac{\pi}{2}+n\pi$$ into one of the above equations to learn what additional conditions $$\displaystyle x$$ and $$\displaystyle y$$ have to satisfy...

dhiab

#### sa-ri-ga-ma

Solve : $$\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}$$
$$\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4}$$
$$\displaystyle cos(x)cos(y) + sin(x)sin(y) = \frac{\sqrt{3}}{2}]$$

$$\displaystyle cos(x-y) = \frac{\sqrt{3}}{2}$$

x - y = π/6 .......(1)

If you subtract the above two equation you will get

cos(x+y) = 0 of x+y = π/2.....(2)

From eq.1 and 2, find x and y.

dhiab

MHF Hall of Honor
Hello dhiab
Solve : $$\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}$$
$$\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4}$$
Using $$\displaystyle \sin x \sin y = \tfrac12\big(\cos(x-y) -\cos(x+y)\big)$$:
$$\displaystyle \sin x\sin y=\frac{\sqrt{3}}{4}$$

$$\displaystyle \Rightarrow \cos(x-y)-\cos(x+y) = \frac{\sqrt3}{2}$$ ... (1)
Similarly:

$$\displaystyle \cos x \cos y =\frac{\sqrt{3}}{4}$$

$$\displaystyle \Rightarrow \cos(x-y)+\cos(x+y) = \frac{\sqrt3}{2}$$ ... (2)

$$\displaystyle \cos(x-y) =\frac{\sqrt3}{2}$$

$$\displaystyle \Rightarrow x-y = 2n\pi\pm\frac{\pi}{6}$$ ... (3)

Subtract (1) and (2):
$$\displaystyle \cos(x+y)=0$$

$$\displaystyle \Rightarrow x+y = 2n\pi\pm\frac{\pi}{2}$$ ...(4)

$$\displaystyle x=2n\pi \pm \frac{\pi}{3}$$
$$\displaystyle \Rightarrow y = 2n\pi\pm\frac{\pi}{6}$$