Find the maximum value of $4 sin x + 3 cos x$ .

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Express $3 cos θ - 4 sin θ$ in the form $R cos (θ + α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ .
Solve the trigonometric equation $2 cos (2 x) = 1$ for $0 \leq x \leq π$ .
Simplify the trigonometric expression: $\frac{s i n ( 2 θ )}{1 + c o s ( 2 θ )}$ .
Given that $tan A = \frac{1}{2}$ and $tan B = \frac{1}{3}$ , find the exact value of the acute angle $(A + B)$ .
Find the maximum and minimum values of the function $f (θ) = 8 - 3 sin (2 θ - \frac{π}{4})$ .
Solve for $θ \in [0, 2 π]$ : $4 sin^{2} θ + 8 cos θ - 7 = 0$ .