Solve with multiple compound angles

1 Solve the equation \(2 \sin \left( \theta + 30 ^ { \circ } \right) + 5 \cos \theta = 2 \sin \theta\) for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

3

1. Given that \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\), find the exact value of \(\tan \theta\) in a form involving surds. You need not simplify your answer.
2. Hence solve the equation \(\sin \left( \theta + 45 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) = 3 \cos \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

3 Solve the equation $$\sin \left( \theta + 45 ^ { \circ } \right) = 2 \cos \left( \theta - 30 ^ { \circ } \right)$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5

1. Given that $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ find the exact value of \(\tan x\).
2. Hence find the exact roots of the equation $$\sin \left( x + \frac { 1 } { 6 } \pi \right) - \sin \left( x - \frac { 1 } { 6 } \pi \right) = \cos \left( x + \frac { 1 } { 3 } \pi \right) - \cos \left( x - \frac { 1 } { 3 } \pi \right)$$ for \(0 \leqslant x \leqslant 2 \pi\).

3．（a）Prove that \(\tan 15 ^ { \circ } = 2 - \sqrt { 3 }\)
（b）Solve，for \(0 \leqslant \theta < 360 ^ { \circ }\) ， $$\sin \left( \theta + 60 ^ { \circ } \right) \sin \left( \theta - 60 ^ { \circ } \right) = ( 1 - \sqrt { } 3 ) \cos ^ { 2 } \theta$$ Figure 1 shows a sketch of the curve \(C\) with equation $$y = \cos x \ln ( \sec x ) , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$ The points \(A\) and \(B\) are maximum points on \(C\) ．
（a）Find the coordinates of \(B\) in terms of e ． The finite region \(R\) lies between \(C\) and the line \(A B\) ．
（b）Show that the area of \(R\) is $$\frac { 2 } { \mathrm { e } } \arccos \left( \frac { 1 } { \mathrm { e } } \right) + 2 \ln \left( \mathrm { e } + \sqrt { } \left( \mathrm { e } ^ { 2 } - 1 \right) \right) - \frac { 4 } { \mathrm { e } } \sqrt { } \left( \mathrm { e } ^ { 2 } - 1 \right) .$$ \(\left[ \arccos x \right.\) is an alternative notation for \(\left. \cos ^ { - 1 } x \right]\)

7. $$\mathrm { f } ( x ) = \left[ 1 + \cos \left( x + \frac { \pi } { 4 } \right) \right] \left[ 1 + \sin \left( x + \frac { \pi } { 4 } \right) \right] , \quad 0 \leqslant x \leqslant 2 \pi$$

1. Show that \(\mathrm { f } ( x )\) may be written in the form $$f ( x ) = \left( \frac { 1 } { \sqrt { 2 } } + \cos x \right) ^ { 2 } , \quad 0 \leqslant x \leqslant 2 \pi$$
2. Find the range of the function \(\mathrm { f } ( x )\). The graph of \(y = \mathrm { f } ( x )\) is shown in Figure 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_426_938_849_591} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}
3. Find the coordinates of all the maximum and minimum points on this curve. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_432_942_1535_589} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The region \(R\), bounded by \(y = 2\) and \(y = \mathrm { f } ( x )\), is shown shaded in Figure 3.
4. Find the area of \(R\).

7. (a) (i) Show that $$\sin ( x + 30 ) ^ { \circ } + \sin ( x - 30 ) ^ { \circ } \equiv a \sin x ^ { \circ }$$ where \(a\) is a constant to be found.
(ii) Hence find the exact value of \(\sin 75 ^ { \circ } + \sin 15 ^ { \circ }\), giving your answer in the form \(b \sqrt { 6 }\).
(b) Solve, for \(0 \leq y \leq 360\), the equation $$2 \cot ^ { 2 } y ^ { \circ } + 5 \operatorname { cosec } y ^ { \circ } + \operatorname { cosec } ^ { 2 } y ^ { \circ } = 0$$