1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc

1. Express \(2\cos\theta + 5\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]
2. Find the maximum and minimum values of \(2\cos\theta + 5\sin\theta\) and the smallest possible value of \(\theta\) for which the maximum occurs. [2]

The temperature \(T\) °C, of an unheated building is modelled using the equation $$T = 15 + 2\cos\left(\frac{\pi t}{12}\right) + 5\sin\left(\frac{\pi t}{12}\right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.

1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
2. Calculate, to the nearest half hour, the times when the temperature is predicted to be 12 °C. [6]

\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 \(3\sin x + 2\cos x\) in the form \(R\sin(x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). Hence find, correct to 2 decimal places, the coordinates of the maximum point on the curve \(y = f(x)\), where $$f(x) = 3\sin x + 2\cos x, \quad 0 < x < \pi.$$ [7]

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 \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin(\theta - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac{1}{2}\pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\). [6]

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 find the range of the function \(f(\theta)\), where $$f(\theta) = 7 + 3\cos\theta + 4\sin\theta \quad \text{for } 0 \leqslant \theta \leqslant 2\pi.$$ Write down the greatest possible value of \(\frac{1}{7 + 3\cos\theta + 4\sin\theta}\). [6]

Express \(3\sin x + 2\cos x\) in the form \(R\sin(x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) Hence find, correct to 2 decimal places, the coordinates of the maximum point on the curve \(y = f(x)\), where $$f(x) = 3\sin x + 2\cos x, \quad 0 \leqslant x \leqslant \pi.$$ [7]

In Fig. 6, OAB is a thin bent rod, with OA = \(a\) metres, AB = \(b\) metres and angle OAB = 120°. The bent rod lies in a vertical plane. OA makes an angle \(\theta\) above the horizontal. The vertical height BD of B above O is \(h\) metres. The horizontal through A meets BD at C and the vertical through A meets OD at E. \includegraphics{figure_6}

1. Find angle BAC in terms of \(\theta\). Hence show that $$h = a\sin\theta + b\sin(\theta - 60°).$$ [3]
2. Hence show that \(h = (a + \frac{1}{2}b)\sin\theta - \frac{\sqrt{3}}{2}b\cos\theta\). [3]

The rod now rotates about O, so that \(\theta\) varies. You may assume that the formulae for \(h\) in parts (i) and (ii) remain valid.

1. Show that OB is horizontal when \(\tan\theta = \frac{\sqrt{3}b}{2a + b}\). [3]

In the case when \(a = 1\) and \(b = 2\), \(h = 2\sin\theta - \sqrt{3}\cos\theta\).

1. Express \(2\sin\theta - \sqrt{3}\cos\theta\) in the form \(R\sin(\theta - \alpha)\). Hence, for this case, write down the maximum value of \(h\) and the corresponding value of \(\theta\). [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]

The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_4}

1. Determine the coordinates of \(B\). [6]
2. Hence determine the exact value of \(\tan \alpha\). [3]

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]

On 18 March 2019 there were 12 hours of daylight in Inverness. On 16 June 2019, 90 days later, there will be 18 hours of daylight in Inverness. Jude decides to model the number of hours of daylight in Inverness, \(N\), by the formula $$N = A + B\sin t°$$ where \(t\) is the number of days after 18 March 2019.

1. 1. State the value that Jude should use for \(A\). [1 mark]
   2. State the value that Jude should use for \(B\). [1 mark]
   3. Using Jude's model, calculate the number of hours of daylight in Inverness on 15 May 2019, 58 days after 18 March 2019. [1 mark]
   4. Using Jude's model, find how many days during 2019 will have at least 17.4 hours of daylight in Inverness. [4 marks]
   5. Explain why Jude's model will become inaccurate for 2020 and future years. [1 mark]
2. Anisa decides to model the number of hours of daylight in Inverness with the formula $$N = A + B\sin \left(\frac{360}{365}t\right)°$$ Explain why Anisa's model is better than Jude's model. [1 mark]