Model and solve problems that involve equations of the form 𝑎 cos(𝑥) + 𝑏 sin(𝑥) = 𝑐.

A lighthouse beam's intensity (lumens) at time \(t\) seconds is modelled by \(I(t) = 3\cos(t) + 4\sin(t)\). What is the maximum intensity of the beam?

A communications tower transmits a radio signal whose power output \(P\) (in kilowatts) varies with time \(t\) (in hours after midnight) according to the equation \[ P = 5\cos(t) + 12\sin(t) \] Determine the times during the first 12 hours (\(0 \le t \le 12\)) when the power output is exactly 10 kilowatts. Express your answers in hours correct to 2 decimal places.

A coastal engineer models the horizontal displacement, \(d\) metres, of a buoy from its anchor point using the equation \(3\cos(\theta) + 4\sin(\theta) = 2.5\), where \(\theta\) is the angle of the prevailing current measured in radians from due east, \(0 \leq \theta < 2\pi\). a) Express \(3\cos(\theta) + 4\sin(\theta)\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). Give the value of \(\alpha\) correct to four decimal places. (2 marks) b) Use your result from part (a) to determine all values of \(\theta\) that satisfy the displacement equation. Give your answers correct to two decimal places. (1 mark)

A tidal buoy records the vertical displacement, \(h\) metres, above mean sea level at time \(t\) hours after midnight. The displacement satisfies the equation \(3\cos(t) + 4\sin(t) = 2.5\) at certain times during the observation period \(0 \leq t \leq 12\). a) Express \(3\cos(t) + 4\sin(t)\) in the form \(R\cos(t - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). State the exact value of \(R\) and the value of \(\alpha\) correct to four decimal places. (2 marks) b) Use your result from part (a) to determine the first time after midnight when the buoy displacement satisfies the given equation. Give your answer in hours, correct to two decimal places. (1 mark)