I am interested in how you would encourage students to layout their working to a trigonometric equation.
For instance, let's consider this problem:
Solve the equation $6\cos x - 8\sin x = 7$ for $0 < x < 2 \pi$
Then I might write $$6 \cos x - 8 \sin x = -10 \sin \left(x - 0.6435... \right)$$
and then solve $$-10 \sin(x - 0.6435...) = 7$$
This re-arranges to $$\sin(x-0.6435...) = -0.7$$ The principal value is $x - 0.6435... = -0.7753...$.
The values in range are $x-0.6435... = \pi + 0.7753...$ and $x - 0.6435... = 2\pi - 0.7753...$
And if you solve these you get $x=4.56$ and $x=6.15$ to two decimal places.
However, this has a LOT of decimals and I find that it is not the clearest way for a student to lay out their working.
Of course we can work more exactly, but then that'd be quite difficult for lower ability high school students to understand rather than the formulaic approach.
So how would you advise a student to layout their working for solving a trigonometric equation to keep it as simple and neat as possible?
Edit: in UK schools, solving equations of the form $a\cos x + b \sin x = c$ is almost always encouraged to be done by first writing $a \cos x + b \sin x$ in the form $R \sin (x + \alpha)$ or $R \cos (x + \alpha)$, depending on what is most suitable. That is why I skipped that step.
My main point is I guess a general question in how do you avoid the excessive use of decimals in a problem like this? Of course, as I said above, you would work exactly and for me and you as skilled mathematicians this is easy. However, for the lower ability students, I wondered if anybody else had a neater way to lay something like this out that doesn't up the demand that much.