Adding sinusoids Problem set B

Write \(7\cos(5t + 1.1) + 8\cos(5t + 4)\) in the form \(A\cos(5t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).

Write \(8\sin(2t - 0.6) + 5\sin(2t + 1.5)\) in the form \(A\sin(2t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).

Use complex exponentials.

1. (a) Show that \(9\sin(t + 1.2) + 4\sin(t + 2.6) = \mathrm{Im}\left(9e^{(t + 1.2)i} + 4e^{(t + 2.6)i}\right)\).
2. (b) Hence write \(9\sin(t + 1.2) + 4\sin(t + 2.6)\) in the form \(A\sin(t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).

Write each expression in the form \(A\cos(x + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).

1. (a) \(9\cos x + 40\sin x\)
2. (b) \(6\cos x - 2\sin x\)

The graph shows \(y = 6\cos x + 2\cos(x - 1.5)\) for \(0 \le x \le 6.5\). P is the maximum, Q is the minimum and R is the right-hand x-intercept.

1. (a) Write \(y\) in the form \(A\cos(x + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).
2. (b) Hence find the coordinates of P, Q and R.

Two generators feed one lamp. At time \(t\) milliseconds the voltages, in volts, are \(V_1 = 14\cos(60t + 0.5)\) and \(V_2 = 9\cos(60t - 1)\).

1. (a) Write the total voltage \(V(t) = V_1 + V_2\) in the form \(A\cos(60t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).
2. (b) Find the greatest voltage.

A paddle boarder is on a lake. At time \(t\) seconds she is lifted by waves from a motor boat, \(H_1(t) = 0.6\sin(3t + 1)\text{ m}\), and from a jet ski, \(H_2(t) = 0.4\sin(3t - 0.5)\text{ m}\).

1. (a) Write the total height \(H(t) = H_1 + H_2\) in the form \(A\sin(3t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).
2. (b) Find the times in the first 6 seconds when \(H(t) = 0\).
3. (c) Find the first time at which the total height is greatest.
4. (d) At that time, what percentage of the wave is due to the motor boat?

A footbridge deck sways from its mean position because of wind and walkers. At time \(t\) seconds, in centimetres, \(D_1 = 2.4\cos(0.8t)\) and \(D_2 = 1.1\cos(0.8t + 1.3)\).

1. (a) Write \(D(t) = D_1 + D_2\) in the form \(A\cos(0.8t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).
2. (b) Find the greatest value of \(D\).
3. (c) Find the first time \(t \ge 0\) at which \(D\) is greatest.

Three lamps shine on a sensor. At time \(t\) seconds the light intensities, in lux, are \(S_1 = 4\cos(3t + 0.5)\), \(S_2 = 6\cos(3t - 1)\) and \(S_3 = 3\cos(3t + 2.5)\).

1. (a) Write \(S(t) = S_1 + S_2 + S_3\) in the form \(A\cos(3t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).
2. (b) State the greatest value of \(S\).

Two seismic waves add at a sensor. At time \(t\) seconds the displacements, in micrometres, are \(S_1 = 9\sin(5t)\) and \(S_2 = 4\sin(5t + 2.4)\).

1. (a) Write \(R(t) = S_1 + S_2\) in the form \(A\sin(5t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).
2. (b) At \(t = 0.05\), what percentage of the signal is due to \(S_1\)?