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!
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 $\displaystyle rcos(\theta-\alpha)$