Express and solve equation

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]

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.

9

1. Show that \(\sin \theta + \sqrt { 3 } \cos \theta\) can be expressed in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). State the values of \(R\) and \(\alpha\).
2. Hence find the value of \(\theta\), where \(0 < \theta < \pi\), such that \(\sin \theta + \sqrt { 3 } \cos \theta = 0.8\).

\includegraphics{figure_7} The diagram shows the curve with equation \(y = \sin 2x + 3\cos 2x\) for \(0 \leqslant x \leqslant \pi\). At the points \(P\) and \(Q\) on the curve, the gradient of the curve is 3.

1. Find an expression for \(\frac{dy}{dx}\). [2]
2. By first expressing \(\frac{dy}{dx}\) in the form \(R\cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), find the \(x\)-coordinates of \(P\) and \(Q\), giving your answers correct to 4 significant figures. [8]

\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) which has a maximum point at \((-45, 7)\) and a minimum point at \((135, -1)\).

1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
   1. \(y = f(|x|)\),
   2. \(y = 1 + 2f(x)\). [6]

Given that $$f(x) = A + 2\sqrt{2} \cos x^{\circ} - 2\sqrt{2} \sin x^{\circ}, \quad x \in \mathbb{R}, \quad -180 \leq x \leq 180,$$ where \(A\) is a constant,

1. show that f(x) can be expressed in the form $$f(x) = A + R \cos (x + \alpha)^{\circ},$$ where \(R > 0\) and \(0 < \alpha < 90\), [3]
2. state the value of \(A\), [1]
3. find, to \(1\) decimal place, the \(x\)-coordinates of the points where the curve \(y = f(x)\) crosses the \(x\)-axis. [4]

\includegraphics{figure_7} The diagram shows the curve \(y = \text{f}(x)\) which has a maximum point at \((-45, 7)\) and a minimum point at \((135, -1)\).

1. Showing the coordinates of any stationary points, sketch the curve with equation \(y = 1 + 2\text{f}(x)\). [3]

Given that $$\text{f}(x) = A + 2\sqrt{2} \cos x° - 2\sqrt{2} \sin x°, \quad x \in \mathbb{R}, \quad -180 \leq x \leq 180,$$ where \(A\) is a constant,

1. show that f\((x)\) can be expressed in the form $$\text{f}(x) = A + R \cos (x + \alpha)°,$$ where \(R > 0\) and \(0 < \alpha < 90\), [3]
2. state the value of \(A\), [1]
3. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \text{f}(x)\) crosses the \(x\)-axis. [4]

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 \(0 \leq \theta \leq 2\pi\). [7]

Express \(\sin \theta - 3 \cos \theta\) in the form \(R \sin (\theta - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and \(0° < \alpha < 90°\). Hence solve the equation \(\sin \theta - 3 \cos \theta = 1\) for \(0° \leqslant \theta \leqslant 360°\). [7]

In this question you must show detailed reasoning. The function f is defined by \(\text{f}(x) = \cos x + \sqrt{3} \sin x\) with domain \(0 \leqslant x \leqslant 2\pi\).

1. Solve the following equations.
   1. \(\text{f}'(x) = 0\) [4]
   2. \(\text{f}''(x) = 0\) [3]
   The diagram shows the graph of the gradient function \(y = \text{f}'(x)\) for the domain \(0 \leqslant x \leqslant 2\pi\). \includegraphics{figure_5}
2. Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points \(A\), \(B\), \(C\) and \(D\). [2]
3. 1. Explain how to use the graph of the gradient function to find the values of \(x\) for which f(x) is increasing. [1]
   2. Using set notation, write down the set of values of \(x\) for which f(x) is increasing in the domain \(0 \leqslant x \leqslant 2\pi\). [2]

A particle \(P\) moves along a straight line in such a way that at time \(t\) seconds \(P\) has velocity \(v\) m 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 4 significant figures. [3]
2. Hence find the two smallest positive values of \(t\) for which \(P\) is moving, in either direction, with a speed of 3 m s\(^{-1}\). [3]

In this question you must show detailed reasoning.

1. Express \(8\cos x + 5\sin x\) in the form \(R\cos(x - \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). [3]
2. Hence solve the equation \(8\cos x + 5\sin x = 6\) for \(0 \leqslant x < 2\pi\), giving your answers correct to 4 decimal places. [3]