Expand then express in harmonic form

A question is this type if and only if it explicitly requires first expanding a compound angle expression (e.g., sin(x-30°)) before expressing the result in harmonic form.

3 questions · Standard +0.3

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CAIE P3 2017 June Q8
8 marks Standard +0.3
8
  1. By first expanding \(2 \sin \left( x - 30 ^ { \circ } \right)\), express \(2 \sin \left( x - 30 ^ { \circ } \right) - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [5]
  2. Hence solve the equation $$2 \sin \left( x - 30 ^ { \circ } \right) - \cos x = 1$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
OCR C3 2008 June Q8
10 marks Standard +0.3
8 The expression \(\mathrm { T } ( \theta )\) is defined for \(\theta\) in degrees by $$\mathrm { T } ( \theta ) = 3 \cos \left( \theta - 60 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) .$$
  1. Express \(\mathrm { T } ( \theta )\) in the form \(A \sin \theta + B \cos \theta\), giving the exact values of the constants \(A\) and \(B\). [3]
  2. Hence express \(\mathrm { T } ( \theta )\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  3. Find the smallest positive value of \(\theta\) such that \(\mathrm { T } ( \theta ) + 1 = 0\).
CAIE P3 2013 June Q6
8 marks Standard +0.3
  1. By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - ( \sqrt { } 2 ) \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - ( \sqrt { } 2 ) \sin x = 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
  3. Express \(\frac { 1 } { x ^ { 2 } ( 2 x + 1 ) }\) in the form \(\frac { A } { x ^ { 2 } } + \frac { B } { x } + \frac { C } { 2 x + 1 }\).
  4. The variables \(x\) and \(y\) satisfy the differential equation $$y = x ^ { 2 } ( 2 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x }$$ and \(y = 1\) when \(x = 1\). Solve the differential equation and find the exact value of \(y\) when \(x = 2\). Give your value of \(y\) in a form not involving logarithms.
    (a) The complex number \(w\) is such that \(\operatorname { Re } w > 0\) and \(w + 3 w ^ { * } = \mathrm { i } w ^ { 2 }\), where \(w ^ { * }\) denotes the complex conjugate of \(w\). Find \(w\), giving your answer in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
    (b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z - 2 i | \leqslant 2\) and \(0 \leqslant \arg ( z + 2 ) \leqslant \frac { 1 } { 4 } \pi\). Calculate the greatest value of \(| z |\) for points in this region, giving your answer correct to 2 decimal places.