Express and solve equation

1. (i) Express \(4 \sin x + 3 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
   (ii) State the minimum value of \(4 \sin x + 3 \cos x\) and the smallest positive value of \(x\) for which this minimum value occurs.
   (iii) Solve the equation

$$4 \sin 2 \theta + 3 \cos 2 \theta = 2$$ for \(\theta\) in the interval \(0 \leq \theta \leq \pi\), giving your answers to 2 decimal places.

7 $$f ( x ) = 2 + \cos x + 3 \sin x$$

1. Express \(\mathrm { f } ( x )\) in the form $$\mathrm { f } ( x ) = a + b \cos ( x - c )$$ where \(a , b\) and \(c\) are constants, \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\).
2. Solve the equation \(\mathrm { f } ( x ) = 0\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\).
3. Use Simpson's rule with four strips, each of width 0.5 , to find an approximate value for $$\int _ { 0 } ^ { 2 } f ( x ) d x$$

6. (i) Express \(3 \cos x ^ { \circ } + \sin x ^ { \circ }\) in the form \(R \cos ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(ii) Using your answer to part (a), or otherwise, solve the equation $$6 \cos ^ { 2 } x ^ { \circ } + \sin 2 x ^ { \circ } = 0$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place where appropriate.

6. (i) Express \(\sqrt { 3 } \sin \theta + \cos \theta\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
(ii) State the maximum value of \(\sqrt { 3 } \sin \theta + \cos \theta\) and the smallest positive value of \(\theta\) for which this maximum value occurs.
(iii) Solve the equation $$\sqrt { 3 } \sin \theta + \cos \theta + \sqrt { 3 } = 0$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\), giving your answers in terms of \(\pi\).

5

1. Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(4 \cos \theta - \sin \theta = 2\), giving all solutions for which \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\).

5

1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(3 \sin \theta + 2 \cos \theta = \frac { 7 } { 2 }\), giving all solutions for which \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

1 Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
Hence solve the equation \(3 \cos \theta + 4 \sin \theta = 2\) for \(- \pi \leqslant \theta \leqslant \pi\).

1 Express \(\sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Hence solve the equation \(\sin \theta - 3 \cos \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6 The function \(\mathrm { f } ( \theta ) = 3 \sin \theta + 4 \cos \theta\) is to be expressed in the form \(r \sin ( \theta + \alpha )\) where \(r > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).

1. Find the values of \(r\) and \(\alpha\).
2. Write down the maximum and minimum value of \(\mathrm { f } ( \theta )\).
3. Solve the equation \(\mathrm { f } ( \theta ) = 1\) for \(0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\).

2 Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < { } _ { 2 } ^ { 1 } \pi\).-
Hence solve the equation \(4 \cos \theta - \sin \theta = 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).

6 Express \(\sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Hence solve the equation \(\sin \theta - 3 \cos \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

2 Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { - \pi } { 2 }\).
Hence solve the equation \(3 \cos \theta + 4 \sin \theta = 2\) for \(\quad - \pi \leqslant \theta \leqslant \pi\).

5 Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
Hence solve the equation \(4 \cos \theta - \sin \theta = 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).

7

1. By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x = 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

4

1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

8

1. Express \(3 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence
   1. solve the equation \(3 \sin \theta + 4 \cos \theta + 1 = 0\), giving all solutions for which \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\),
   2. find the values of the positive constants \(k\) and \(c\) such that $$- 37 \leqslant k ( 3 \sin \theta + 4 \cos \theta ) + c \leqslant 43$$ for all values of \(\theta\).

6 \includegraphics[max width=\textwidth, alt={}, center]{00a4be37-c095-4d9c-a1cd-d03b8ab1d411-3_553_579_274_726} The diagram shows the curve \(y = 8 \sin ^ { - 1 } \left( x - \frac { 3 } { 2 } \right)\). The end-points \(A\) and \(B\) of the curve have coordinates ( \(a , - 4 \pi\) ) and ( \(b , 4 \pi\) ) respectively.

1. State the values of \(a\) and \(b\).
2. It is required to find the root of the equation \(8 \sin ^ { - 1 } \left( x - \frac { 3 } { 2 } \right) = x\).
   1. Show by calculation that the root lies between 1.7 and 1.8.
   2. In order to find the root, the iterative formula $$x _ { n + 1 } = p + \sin \left( q x _ { n } \right) ,$$ with a suitable starting value, is to be used. Determine the values of the constants \(p\) and \(q\) and hence find the root correct to 4 significant figures. Show the result of each step of the iteration process.

1. (i) Solve \(0 \leq \theta \leq 180 ^ { 0 }\), the equation

$$4 \cos \theta = \sqrt { 3 } \operatorname { cosec } \theta$$ (ii) Solve, for \(0 \leq x \leq 2 \pi\), the equation $$\cos x - \sqrt { 3 } \sin x = \sqrt { 3 }$$

1. (i) Solve, for \(0 \leqslant x < \frac { \pi } { 2 }\), the equation

$$4 \sin x = \sec x$$ (ii) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$5 \sin \theta - 5 \cos \theta = 2$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

8 In this question you must show detailed reasoning.

1. Express \(\cos x + \sqrt { 3 } \sin x\) in the form \(\mathrm { R } \sin ( \mathrm { x } + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the values of \(R\) and \(\alpha\) in exact form.
2. Hence solve the equation \(\cos x = \sqrt { 3 } ( 1 - \sin x )\) for values of \(x\) in the interval \(- \pi \leqslant x \leqslant \pi\). Give the roots of this equation in exact form.

5. (a) Express \(3 \cos x ^ { \circ } + \sin x ^ { \circ }\) in the form \(R \cos ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(b) Using your answer to part (a), or otherwise, solve the equation $$6 \cos ^ { 2 } x ^ { \circ } + \sin 2 x ^ { \circ } = 0$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place where appropriate.

7. (a) Express \(4 \sin x + 3 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
(b) State the minimum value of \(4 \sin x + 3 \cos x\) and the smallest positive value of \(x\) for which this minimum value occurs.
(c) Solve the equation $$4 \sin 2 \theta + 3 \cos 2 \theta = 2$$ for \(\theta\) in the interval \(0 \leq \theta \leq \pi\), giving your answers to 2 decimal places.

5. (a) Express \(\sqrt { 3 } \sin \theta + \cos \theta\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
(b) State the maximum value of \(\sqrt { 3 } \sin \theta + \cos \theta\) and the smallest positive value of \(\theta\) for which this maximum value occurs.
(c) Solve the equation $$\sqrt { 3 } \sin \theta + \cos \theta + \sqrt { 3 } = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\), giving your answers in terms of \(\pi\).

2

1. Express \(\sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. Hence find the values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) for which $$\sin x - 3 \cos x + 2 = 0$$ giving your values of \(x\) to the nearest degree.

5

1. 1. Express \(3 \sin x + 4 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(3 \sin 2 \theta + 4 \cos 2 \theta = 5\) in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your solutions to the nearest \(0.1 ^ { \circ }\).
2. 1. Show that the equation \(\tan 2 \theta \tan \theta = 2\) can be written as \(2 \tan ^ { 2 } \theta = 1\).
   2. Hence solve the equation \(\tan 2 \theta \tan \theta = 2\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), giving your solutions to the nearest \(0.1 ^ { \circ }\).
3. 1. Use the Factor Theorem to show that \(2 x - 1\) is a factor of \(8 x ^ { 3 } - 4 x + 1\).
   2. Show that \(4 \cos 2 \theta \cos \theta + 1\) can be written as \(8 x ^ { 3 } - 4 x + 1\) where \(x = \cos \theta\).
   3. Given that \(\theta = 72 ^ { \circ }\) is a solution of \(4 \cos 2 \theta \cos \theta + 1 = 0\), use the results from parts (c)(i) and (c)(ii) to show that the exact value of \(\cos 72 ^ { \circ }\) is \(\frac { ( \sqrt { 5 } - 1 ) } { p }\) where \(p\) is an integer.
      [0pt] [3 marks]