Question

We already know that R = 5 and α = 37°

Answer 1

(b)

Given the equation:

$8\sin (x)+6\cos (x)=3$

But this is equivalent to:

$4\sin (x)+3\cos (x)=\frac{3}{2}$

Then, using the result on (a):

$\begin{gathered} 5\sin (x+37\degree)=\frac{3}{2} \\ \Rightarrow\sin (x+37\degree)=\frac{3}{10} \end{gathered}$

Taking the arcsine:

$\begin{gathered} \arcsin (\sin (x+37\degree))=\arcsin (\frac{3}{10}) \\ x+37\degree=\arcsin (\frac{3}{10}) \\ \Rightarrow x=\arcsin (\frac{3}{10})-37\degree \end{gathered}$

The sine is positive in the first and the second quadrant, then:

$\begin{gathered} \arcsin (\frac{3}{10})\approx17.46\degree \\ or \\ \arcsin (\frac{3}{10})\approx180\degree-17.46\degree=162.54\degree \\ or \\ \arcsin (\frac{3}{10})\approx360\degree+17.46\degree=377.46\degree\text{ (this is equivalent to 17.46\degree)} \end{gathered}$

And the solutions are:

$\begin{gathered} x_1\approx162.54\degree-37\degree\Rightarrow x_1\approx125.54\degree^{} \\ x_2\approx377.46\degree-37\degree\Rightarrow x_2\approx340.46\degree \end{gathered}$