regatamin2

2022-01-03

Ive

John Koga

Hint:
$\mathrm{tan}\varphi =\frac{a}{b}$
$\frac{\mathrm{sin}\varphi }{a}=\frac{\mathrm{cos}\varphi }{b}=±\sqrt{\frac{?}{{a}^{2}+{b}^{2}}}$
Now replace the values of a,b

censoratojk

HINT
Begin by assuming that
$a\mathrm{cos}\theta +b\mathrm{sin}\theta$
can be written in the form
$R\mathrm{sin}\left(x+\varphi \right)$
for some constants R and $\varphi$. Expand the second expression using the angle addition formula, form some equations, and you should have it.

Vasquez

$\begin{array}{}\mathrm{tan}\varphi =\frac{a}{b}\\ Thus,\\ \mathrm{cos}\varphi =\frac{b}{\sqrt{{a}^{2}+{b}^{2}}}\\ \mathrm{sin}\varphi =\frac{a}{\sqrt{{a}^{2}+{b}^{2}}}\\ L.H.S.\\ a\mathrm{cos}\theta +b\mathrm{sin}\theta =\sqrt{{a}^{2}+{b}^{2}}\frac{\left(a\mathrm{cos}\theta +b\mathrm{sin}\theta \right)}{\sqrt{{a}^{2}+{b}^{2}}}\\ =\sqrt{{a}^{2}+{b}^{2}}\left(\frac{a}{\sqrt{{a}^{2}+{b}^{2}}}\mathrm{cos}\theta +\frac{b}{\sqrt{{a}^{2}+{b}^{2}}}\mathrm{sin}\theta \right)\end{array}$