Express in harmonic form

1 Express \(\cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Hence show that the equation \(\cos \theta - 3 \sin \theta = 4\) has no solution.

1 Express \(\cos \theta + \sqrt { 3 } \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are exact values to be determined.

2 Given that $$6 \cos \theta + 8 \sin \theta \equiv R \cos ( \theta + \alpha )$$ find the value of \(R\). Circle your answer. 681014

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17 2 Given that $$6 \cos \theta + 8 \sin \theta \equiv R \cos ( \theta + \alpha )$$ find the value of \(R\). Circle your answer. 681014 3 Determine which one of these graphs does not represent \(y\) as a function of \(x\). Tick \(( \checkmark )\) one box. \includegraphics[max width=\textwidth, alt={}, center]{076ea8e9-9295-46d2-b5f9-b27fa969129e-03_2246_974_443_495} 4

1. Use the factor theorem to prove that \(x - 6\) is a factor of \(\mathrm { p } ( x )\). \(4 \quad \mathrm { p } ( x ) = 4 x ^ { 3 } - 15 x ^ { 2 } - 48 x - 36\) 4
2. 1. Prove that the graph of \(y = \mathrm { p } ( x )\) intersects the \(x\)-axis at exactly one point.
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3. (ii) State the coordinates of this point of intersection.