sin(x)/sin(y)= 1/2 cos(x)/cos(y)= 3

prove sin(x+y) = (7/3)sin(x)cos(x)

I dont know what to do with the fractions, do i solve for one and then try and put it in the equation or something?

like sin(x)= (1/2)sin(y)

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- Apr 3rd 2010, 09:42 AM #1

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## another question!

sin(x)/sin(y)= 1/2 cos(x)/cos(y)= 3

prove sin(x+y) = (7/3)sin(x)cos(x)

I dont know what to do with the fractions, do i solve for one and then try and put it in the equation or something?

like sin(x)= (1/2)sin(y)

- Apr 3rd 2010, 10:15 AM #2
Um, yes I think so.

like sin(x)= (1/2)sin(y)

- Apr 3rd 2010, 12:21 PM #3

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Hello, OmegaCenturion!

Given: .$\displaystyle \begin{Bmatrix}\dfrac{\sin x}{\sin y} &=& \frac{1}{2} & [1] \\ \\[-3mm] \dfrac{\cos x}{\cos y}&=& 3 & [2] \end{Bmatrix}$

Prove: .$\displaystyle \sin(x+y) \:=\: \tfrac{7}{3}\sin x \cos x$

From [1] we have: .$\displaystyle \sin y \:=\:2\sin x$ .(a)

From [2] we have: .$\displaystyle \cos y \:=\:\tfrac{1}{3}\cos x$ .(b)

We know that: .$\displaystyle \sin(x+y) \;=\;\sin x\cos y + \cos x\sin y$ .(c)

Substitute (a) and (b) into (c):

. . $\displaystyle \sin(x+y) \;=\;\sin x\left(\tfrac{1}{3}\cos x\right) + \cos x\left(2\sin x\right)$

. . . . . . . . .$\displaystyle =\; \tfrac{1}{3}\sin x\cos x + 2\sin x\cos x$

. . . . . . . . .$\displaystyle =\;\tfrac{7}{3}\sin x\cos x$