1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc

242 questions

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CAIE P2 2020 June Q6
10 marks Standard +0.8
6
  1. Prove that $$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) \equiv \sqrt { 8 } \cos \left( \theta + \frac { 1 } { 4 } \pi \right)$$
  2. Solve the equation $$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) = 1$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\). Give the answer correct to 3 significant figures.
  3. Find \(\int \sin x \left( \operatorname { cosec } \frac { 1 } { 2 } x - \sec \frac { 1 } { 2 } x \right) \mathrm { d } x\).
CAIE P2 2022 June Q8
9 marks Challenging +1.2
8
  1. Express \(3 \sin 2 \theta \sec \theta + 10 \cos \left( \theta - 30 ^ { \circ } \right)\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(3 \sin 4 \beta \sec 2 \beta + 10 \cos \left( 2 \beta - 30 ^ { \circ } \right) = 2\) for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2023 June Q7
11 marks Standard +0.3
7
  1. Express \(7 \cos \theta + 24 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Solve the equation \(7 \cos \theta + 24 \sin \theta = 18\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  3. As \(\beta\) varies, the greatest possible value of $$\frac { 150 } { 7 \cos \frac { 1 } { 2 } \beta + 24 \sin \frac { 1 } { 2 } \beta + 50 }$$ is denoted by \(V\).
    Find the value of \(V\) and determine the smallest positive value of \(\beta\) (in degrees) for which the value of \(V\) occurs.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2021 March Q7
9 marks Standard +0.8
7
  1. Express \(5 \sqrt { 3 } \cos x + 5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  2. As \(x\) varies, find the least possible value of $$4 + 5 \sqrt { 3 } \cos x + 5 \sin x$$ and determine the corresponding value of \(x\) where \(- \pi < x < \pi\).
  3. Find \(\int \frac { 1 } { ( 5 \sqrt { 3 } \cos 3 \theta + 5 \sin 3 \theta ) ^ { 2 } } d \theta\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2020 November Q7
10 marks Standard +0.3
7 A curve is defined by the parametric equations $$x = 3 t - 2 \sin t , \quad y = 5 t + 4 \cos t$$ where \(0 \leqslant t \leqslant 2 \pi\). At each of the points \(P\) and \(Q\) on the curve, the gradient of the curve is \(\frac { 5 } { 2 }\).
  1. Show that the values of \(t\) at \(P\) and \(Q\) satisfy the equation \(10 \cos t - 8 \sin t = 5\).
  2. Express \(10 \cos t - 8 \sin t\) in the form \(R \cos ( t + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
  3. Hence find the values of \(t\) at the points \(P\) and \(Q\).
CAIE P2 2022 November Q8
10 marks Standard +0.3
8 The expression \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = 12 \sin \theta \cos \theta + 16 \cos ^ { 2 } \theta\).
  1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( 2 \theta - \alpha ) + k\), where \(R > 0,0 < \alpha < \frac { 1 } { 2 } \pi\) and \(k\) is a constant. State the values of \(R\) and \(k\), and give the value of \(\alpha\) correct to 4 significant figures.
  2. Find the smallest positive value of \(\theta\) satisfying the equation \(\mathrm { f } ( \theta ) = 17\).
  3. Find \(\int f ( \theta ) d \theta\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2024 November Q7
9 marks Standard +0.3
7
  1. Express \(4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right)\) in the form $$a + R \sin ( 2 \theta - \alpha ) ,$$ where \(a\) and \(R\) are positive integers and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-13_2723_33_99_21}
  2. Hence find the smallest positive value of \(\theta\) satisfying the equation $$\frac { 1 } { 5 } + 4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right) = 0 .$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-14_2714_38_109_2010}
CAIE P2 2002 June Q4
8 marks Moderate -0.3
4
  1. Express \(3 \cos \theta + 2 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 1 decimal place.
  2. Solve the equation $$3 \cos \theta + 2 \sin \theta = 3.5$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
  3. The graph of \(y = 3 \cos \theta + 2 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), has one stationary point. State the coordinates of this point. \includegraphics[max width=\textwidth, alt={}, center]{9b103197-7ba0-427a-b983-34edb51b6cca-3_421_823_299_662} The diagram shows the curve \(y = 2 x \mathrm { e } ^ { - x }\) and its maximum point \(P\). Each of the two points \(Q\) and \(R\) on the curve has \(y\)-coordinate equal to \(\frac { 1 } { 2 }\).
CAIE P2 2004 June Q4
8 marks Moderate -0.3
4
  1. Express \(3 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$3 \sin \theta + 4 \cos \theta = 4.5$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), correct to 1 decimal place.
  3. Write down the least value of \(3 \sin \theta + 4 \cos \theta + 7\) as \(\theta\) varies.
CAIE P2 2008 June Q5
7 marks Moderate -0.3
5
  1. Express \(5 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$5 \cos \theta - \sin \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2012 June Q4
8 marks Standard +0.3
4
  1. Express \(9 \sin \theta - 12 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places. Hence
  2. solve the equation \(9 \sin \theta - 12 \cos \theta = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\),
  3. state the largest value of \(k\) for which the equation \(9 \sin \theta - 12 \cos \theta = k\) has any solutions.
CAIE P2 2013 June Q7
10 marks Standard +0.3
7
  1. Express \(5 \sin 2 \theta + 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. Hence
  2. solve the equation $$5 \sin 2 \theta + 2 \cos 2 \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\),
  3. determine the least value of \(\frac { 1 } { ( 10 \sin 2 \theta + 4 \cos 2 \theta ) ^ { 2 } }\) as \(\theta\) varies.
CAIE P2 2017 June Q5
7 marks Moderate -0.3
5
  1. Express \(2 \cos \theta + ( \sqrt { } 5 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(2 \cos \theta + ( \sqrt { } 5 ) \sin \theta = 1\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2018 June Q7
11 marks Challenging +1.2
7
  1. Express \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
  2. Using your answer from part (i), solve the equation $$5 \cot \theta - 4 \operatorname { cosec } \theta = 2$$ for \(0 < \theta < 2 \pi\).
  3. Find \(\int \frac { 1 } { \left( 5 \cos \frac { 1 } { 2 } x - 2 \sin \frac { 1 } { 2 } x \right) ^ { 2 } } \mathrm {~d} x\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2007 June Q5
7 marks Standard +0.3
5
  1. Express \(\cos \theta + ( \sqrt { } 3 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( \cos \theta + ( \sqrt { } 3 ) \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { 1 } { \sqrt { } 3 }\).
CAIE P3 2013 June Q9
10 marks Standard +0.3
9
  1. Express \(4 \cos \theta + 3 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
  2. Hence
    1. solve the equation \(4 \cos \theta + 3 \sin \theta = 2\) for \(0 < \theta < 2 \pi\),
    2. find \(\int \frac { 50 } { ( 4 \cos \theta + 3 \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).
CAIE P3 2013 June Q4
7 marks Standard +0.3
4
  1. Express \(( \sqrt { } 3 ) \cos x + \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
  2. Hence show that $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( ( \sqrt { } 3 ) \cos x + \sin x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \sqrt { } 3$$
CAIE P3 2015 June Q4
6 marks Moderate -0.3
4
  1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$3 \sin \theta + 2 \cos \theta = 1$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2016 June Q3
6 marks Standard +0.3
3
  1. Express \(( \sqrt { } 5 ) \cos x + 2 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$( \sqrt { } 5 ) \cos \frac { 1 } { 2 } x + 2 \sin \frac { 1 } { 2 } x = 1.2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
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 \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2017 March Q4
7 marks Standard +0.3
4
  1. Express \(8 \cos \theta - 15 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$8 \cos 2 x - 15 \sin 2 x = 4$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2002 November Q5
8 marks Standard +0.3
5
  1. Express \(4 \sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the value of \(\alpha\) correct to 2 decimal places. Hence
  2. solve the equation $$4 \sin \theta - 3 \cos \theta = 2$$ giving all values of \(\theta\) such that \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
  3. write down the greatest value of \(\frac { 1 } { 4 \sin \theta - 3 \cos \theta + 6 }\).
CAIE P3 2005 November Q5
7 marks Moderate -0.3
5 By expressing \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), solve the equation $$8 \sin \theta - 6 \cos \theta = 7$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2008 November Q6
8 marks Standard +0.3
6
  1. Express \(5 \sin x + 12 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$5 \sin 2 \theta + 12 \cos 2 \theta = 11$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2010 November Q8
9 marks Standard +0.3
8
  1. Express \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence, in each of the following cases, find the smallest positive angle \(\theta\) which satisfies the equation
    1. \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta = - 4\),
    2. \(( \sqrt { } 6 ) \cos \frac { 1 } { 2 } \theta + ( \sqrt { } 10 ) \sin \frac { 1 } { 2 } \theta = 3\).