Express double angle or product

8 The expression \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = 12 \sin \theta \cos \theta + 16 \cos ^ { 2 } \theta\).

1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( 2 \theta - \alpha ) + k\), where \(R > 0,0 < \alpha < \frac { 1 } { 2 } \pi\) and \(k\) is a constant. State the values of \(R\) and \(k\), and give the value of \(\alpha\) correct to 4 significant figures.
2. Find the smallest positive value of \(\theta\) satisfying the equation \(\mathrm { f } ( \theta ) = 17\).
3. Find \(\int f ( \theta ) d \theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Express \(4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right)\) in the form $$a + R \sin ( 2 \theta - \alpha ) ,$$ where \(a\) and \(R\) are positive integers and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-13_2723_33_99_21}
2. Hence find the smallest positive value of \(\theta\) satisfying the equation $$\frac { 1 } { 5 } + 4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right) = 0 .$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-14_2714_38_109_2010}

1. (a) Express \(7 \sin 2 \theta - 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
   (b) Hence solve, for \(0 \leqslant \theta < 90 ^ { \circ }\), the equation

$$7 \sin 2 \theta - 2 \cos 2 \theta = 4$$ giving your answers in degrees to one decimal place.
(c) Express \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\) in the form \(a \sin 2 \theta + b \cos 2 \theta + c\), where \(a\), \(b\) and \(c\) are constants to be found.
(d) Use your answers to part (a) and part (c) to deduce the exact maximum value of \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\)

$$f ( x ) = 7 \cos 2 x - 24 \sin 2 x$$ Given that \(\mathrm { f } ( x ) = R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),

1. find the value of \(R\) and the value of \(\alpha\).
2. Hence solve the equation $$7 \cos 2 x - 24 \sin 2 x = 12.5$$ for \(0 \leqslant x < 180 ^ { \circ }\), giving your answers to 1 decimal place.
3. Express \(14 \cos ^ { 2 } x - 48 \sin x \cos x\) in the form \(a \cos 2 x + b \sin 2 x + c\), where \(a , b\), and \(c\) are constants to be found.
4. Hence, using your answers to parts (a) and (c), deduce the maximum value of $$14 \cos ^ { 2 } x - 48 \sin x \cos x$$