I need a miracle! Can someone pls help me! I need to noe WHEN and HOW to use the R-formulae! and WHAT the R-formuale is in the first place! THANKX!

2. The R-formula is used when you have $\displaystyle cos\theta \pm sin\theta~=~a~$ where a is a non-zero value.

Example;

$\displaystyle 3sin\theta+4cos\theta~=~1$

Firstly set

$\displaystyle 3sin\theta+4cos\theta~=~rsin(\theta+\alpha)$

$\displaystyle 3sin\theta+4cos\theta~=~r(sin\theta \cdot cos\alpha+cos\theta \cdot sin\alpha)$

$\displaystyle sin\theta~terms: 3~=~rcos\alpha~\implies 9~=~r^2cos^2\alpha~~~[1]$

$\displaystyle cos\theta~terms: 4~=~rsin\alpha~\implies 16~=~r^2sin^2\alpha~~~[2]$

$\displaystyle [1]+[2]\rightarrow~25~=~r^2(cos^2\alpha+sin^2\alpha)$

$\displaystyle \implies~25~=~r^2$

$\displaystyle \implies~r~=~5$

Hence,

$\displaystyle 5cos\alpha~=~3~~~~~,~~~~~5sin\alpha~=~4$

$\displaystyle cos\alpha~=~\frac{3}{5}~~~~~,~~~~~sin\alpha~=~\fra c{4}{5}$

$\displaystyle \implies~\alpha~\approx~53.13^o$

Thus,

$\displaystyle 3sin\theta+4cos\theta~=~5sin(\theta+53.13^o)~=~1$

The same technique is also used with $\displaystyle rsin(\theta-\alpha)~,~rcos(\theta+\alpha)$ and $\displaystyle rcos(\theta-\alpha)$

3. Ahhhh. That's what the "R-formula" is supposed to be. I didn't know it had an official name.

-Dan