# Thread: Find FI in: A = B.sin(FI) + C.cos(FI)

1. ## Find FI in: A = B.sin(FI) + C.cos(FI)

Hello all,

I need to solve this problem. Can anyone give me some directions to follow or some reference to some article? Thanks in advance.

The problem is: I need to find the angle FI in the following equation:

A = B * sin(FI) + C * cos(FI)

Best regards to all,
Emmanuel.

2. Hello Emmanuel

Welcome to Math Help Forum!
Originally Posted by Emmanuel
Hello all,

I need to solve this problem. Can anyone give me some directions to follow or some reference to some article? Thanks in advance.

The problem is: I need to find the angle FI in the following equation:

A = B * sin(FI) + C * cos(FI)

Best regards to all,
Emmanuel.
Let $\displaystyle b\sin\phi + c\cos\phi = r \sin(\phi+\alpha) = r\sin\phi\cos\alpha+r\cos\phi\sin\alpha$

Compare coefficients:

$\displaystyle r\cos\alpha = b$ (1)

$\displaystyle r\sin\alpha = c$ (2)

$\displaystyle r^2(\cos^2\alpha+\sin^2\alpha) = b^2+c^2$

$\displaystyle \Rightarrow r = \sqrt{b^2 + c^2}$. This gives the value of $\displaystyle r$.

Then divide (2) by (1) to get $\displaystyle \tan\alpha = \tfrac{c}{b}$, which gives the value of $\displaystyle \alpha$.

Finally, $\displaystyle r\sin(\phi+\alpha) = a \Rightarrow \phi = \arcsin(\tfrac{a}{r})-\alpha$