Adding sinusoids Problem set A

Write \(9\cos(4t + 0.3) + 5\cos(4t + 1.9)\) in the form \(A\cos(4t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).

Write \(11\sin(t + 0.7) + 6\sin(t + 2.9)\) in the form \(A\sin(t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).

Use complex exponentials.

1. (a) Show that \(6\sin(3t + 2) + 2\sin(3t + 0.5) = \mathrm{Im}\left(6e^{(3t + 2)i} + 2e^{(3t + 0.5)i}\right)\).
2. (b) Hence write \(6\sin(3t + 2) + 2\sin(3t + 0.5)\) in the form \(A\sin(3t + B)\), 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) \(12\cos x + 5\sin x\)
2. (b) \(4\cos x - 7\sin x\)

The graph shows \(y = 4\cos x + 3\cos(x - 0.9)\) 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 voltage sources feed one circuit. At time \(t\) seconds the voltages, in volts, are \(V_1 = 15\cos(40t)\) and \(V_2 = 10\cos(40t + 2)\).

1. (a) Write the total voltage \(V(t) = V_1 + V_2\) in the form \(A\cos(40t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).
2. (b) Find the greatest voltage.

A buoy floats near a harbour wall. At time \(t\) seconds it is lifted by waves from a ferry, \(H_1(t) = 0.5\sin(2t)\text{ m}\), and from a fishing boat, \(H_2(t) = 0.3\sin(2t + 1.5)\text{ m}\).

1. (a) Write the total height \(H(t) = H_1 + H_2\) in the form \(A\sin(2t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).
2. (b) Find the times in the first 10 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 ferry?

The water level in a bay, measured from its mean level, is the sum of two tidal effects. At time \(t\) hours, in metres, \(D_1 = 1.8\cos(0.5t)\) and \(D_2 = 0.9\cos(0.5t + 2)\).

1. (a) Write \(D(t) = D_1 + D_2\) in the form \(A\cos(0.5t + B)\), 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 sound sources reach a microphone. At time \(t\) seconds the pressures, in pascals, are \(S_1 = 5\cos(2t)\), \(S_2 = 3\cos(2t + 1)\) and \(S_3 = 2\cos(2t - 2)\).

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

Two radio signals add at a receiver. At time \(t\) milliseconds the strengths, in millivolts, are \(S_1 = 8\sin(7t)\) and \(S_2 = 5\sin(7t + 2)\).

1. (a) Write \(R(t) = S_1 + S_2\) in the form \(A\sin(7t + B)\), where \(A > 0\) and \(-\pi < B \le \pi\).
2. (b) At \(t = 0.1\), what percentage of the signal is due to \(S_1\)?