Question is as follows:-
Hence solve for t for values in the range 0 ≤ t ≤ 2 π rad:
5.5 Cos t + 7.8 Sin t = 4.5
Any help would be greatful, thanks
Well it's a bit ugly but one approach would be to recall that $\displaystyle cos(t) = \sqrt{1 - sin^2(t)}$ and plug that in...
$\displaystyle 5.5~\sqrt{1 - sin^2(t)} + 7.8~sin(t) = 4.5$
Isolate the radical and square it. This will give you a quadratic in sin(t) which you can solve using the quadratic formula. Check for extra, but not valid, solutions.
Also since in reality $\displaystyle cos(t) = \pm \sqrt{1 - sin^2(t)}$ you should also work through the negative solution as well. And again check your solutions with the original equation.
-Dan
Another approach would be to use a linear combination identity to write the equation as:
$\displaystyle \sqrt{5.5^2+7.8^2}\sin\left(t+\tan^{-1}\left(\frac{5.5}{7.8} \right) \right)=4.5$
$\displaystyle \sin\left(t+\tan^{-1}\left(\frac{55}{78} \right) \right)=\frac{45}{\sqrt{9109}}$
Now, after finding the quadrant IV solution, use the identity $\displaystyle \sin(\pi-x)=\sin(x)$ to get the quadrant III solution.