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)=(3)/(2)`

Then, using the result on (a):

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

Taking the arcsine:

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

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

`\begin{gathered} \arcsin ((3)/(10))\approx17.46\degree \\ or \\ \arcsin ((3)/(10))\approx180\degree-17.46\degree=162.54\degree \\ or \\ \arcsin ((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}`