Hello jrr
Welcome to Math Help Forum! Originally Posted by
jrr I have an identily of the form:
cos(a1 – b)/cos(a2 – b) =(x1/x2)*cos(a1)/cos(a2)
Given a1,a2,x1 and x2, how can I solve it for b?
Thanks in advance for any help
It's not clear what the symbols are between the $\displaystyle \cos$ and the brackets, so I shall ignore them for now. (If it turns out to be a power, like $\displaystyle \cos^2(a_1-b)$ the same method applies.)
Expand each of the cosine expressions on the LHS using $\displaystyle \cos(A-B) = \cos A \cos B + \sin A\sin B$, and then divide top and bottom by $\displaystyle \cos b$. Then solve the resulting equation for $\displaystyle \tan b$.
Here's the start: $\displaystyle \frac{\cos(a_1-b)}{\cos(a_2-b)}=\frac{x_1}{x_2}\cdot\frac{\cos a_1}{\cos a_2}$
$\displaystyle \Rightarrow\frac{\cos a_1\cos b+\sin a_1\sin b}{\cos a_2\cos b+\sin a_2\sin b}=\frac{x_1}{x_2}\cdot\frac{\cos a_1}{\cos a_2}$
$\displaystyle \Rightarrow\frac{\cos a_1+\sin a_1\tan b}{\cos a_2+\sin a_2\tan b}=\frac{x_1}{x_2}\cdot\frac{\cos a_1}{\cos a_2}$
Can you now make $\displaystyle \tan b$ the subject of this equation, and hence find $\displaystyle b$?
Grandad