Express and solve equation

7. (i) (a) Express \(( 12 \cos \theta - 5 \sin \theta )\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Hence solve the equation $$12 \cos \theta - 5 \sin \theta = 4$$ for \(0 < \theta < 90 ^ { \circ }\), giving your answer to 1 decimal place.
(ii) Solve $$8 \cot \theta - 3 \tan \theta = 2$$ for \(0 < \theta < 90 ^ { \circ }\), giving your answer to 1 decimal place.

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 \(\pi\).

5

1. 1. Show that the equation \(3 \cos 2 x + 2 \sin x + 1 = 0\) can be written in the form $$3 \sin ^ { 2 } x - \sin x - 2 = 0$$
   2. Hence, given that \(3 \cos 2 x + 2 \sin x + 1 = 0\), find the possible values of \(\sin x\).
2. 1. Express \(3 \cos 2 x + 2 \sin 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation $$3 \cos 2 x + 2 \sin 2 x + 1 = 0$$ for all solutions in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\), giving \(x\) to the nearest \(0.1 ^ { \circ }\).
      (3 marks) \(6 \quad\) A curve has equation \(x ^ { 3 } y + \cos ( \pi y ) = 7\).
3. Find the exact value of the \(x\)-coordinate at the point on the curve where \(y = 1\).
4. Find the gradient of the curve at the point where \(y = 1\).

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]

2

1. Express \(2 \cos x - 5 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\), giving your value of \(\alpha\), in radians, to three significant figures.
2. 1. Hence find the value of \(x\) in the interval \(0 < x < 2 \pi\) for which \(2 \cos x - 5 \sin x\) has its maximum value. Give your value of \(x\) to three significant figures.
   2. Use your answer to part (a) to solve the equation \(2 \cos x - 5 \sin x + 1 = 0\) in the interval \(0 < x < 2 \pi\), giving your solutions to three significant figures.
      [0pt] [3 marks]

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 \(0 ^ { \circ } < \theta < 360 ^ { \circ }\). \section*{June 2005}

7

1. 1. Express \(6 \sin \theta + 8 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value for \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(6 \sin 2 x + 8 \cos 2 x = 7\), giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
2. 1. Prove the identity \(\frac { \sin 2 x } { 1 - \cos 2 x } = \frac { 1 } { \tan x }\).
   2. Hence solve the equation $$\frac { \sin 2 x } { 1 - \cos 2 x } = \tan x$$ giving all solutions in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

2

1. Express \(\cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\), in radians, to three decimal places.
2. 1. Hence write down the minimum value of \(\cos x + 3 \sin x\).
   2. Find the value of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) at which this minimum occurs, giving your answer, in radians, to three decimal places.
3. Solve the equation \(\cos x + 3 \sin x = 2\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving all solutions, in radians, to three decimal places.

1

1. Express \(2 \sin x + \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R\) is a positive constant and \(\alpha\) is an acute angle. Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. Solve the equation \(2 \sin x + \cos x = 1\) for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).

3

1. Express \(4 \cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 360 ^ { \circ }\), giving your value for \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. Hence solve the equation \(4 \cos x + 3 \sin x = 2\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\), giving all solutions to the nearest \(0.1 ^ { \circ }\).
3. Write down the minimum value of \(4 \cos x + 3 \sin x\) and find the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) at which this minimum value occurs.

6

1. 1. Show that the equation \(3 \cos 2 x + 7 \cos x + 5 = 0\) can be written in the form \(a \cos ^ { 2 } x + b \cos x + c = 0\), where \(a , b\) and \(c\) are integers.
   2. Hence find the possible values of \(\cos x\).
2. 1. Express \(7 \sin \theta + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(\alpha\) is an acute angle. Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(7 \sin \theta + 3 \cos \theta = 4\) for all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving \(\theta\) to the nearest \(0.1 ^ { \circ }\).
3. 1. Given that \(\beta\) is an acute angle and that \(\tan \beta = 2 \sqrt { 2 }\), show that \(\cos \beta = \frac { 1 } { 3 }\).
   2. Hence show that \(\sin 2 \beta = p \sqrt { 2 }\), where \(p\) is a rational number.

5 A particle \(P\) moves along a straight line in such a way that at time \(t\) seconds \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 12 \cos t + 5 \sin t .$$

1. Express \(v\) in the form \(R \cos ( t - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to \(\mathbf { 4 }\) significant figures.
2. Hence find the two smallest positive values of \(t\) for which \(P\) is moving, in either direction, with a speed of \(3 \mathrm {~ms} ^ { - 1 }\).