Express and solve equation

A question is this type if and only if it requires expressing in harmonic form and then using that form to solve a trigonometric equation for specific values of the variable.

100 questions · Standard +0.1

1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals
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CAIE P2 2021 June Q3
6 marks Standard +0.3
3 Solve the equation \(\sin \left( 2 \theta + 30 ^ { \circ } \right) = 5 \cos \left( 2 \theta + 60 ^ { \circ } \right)\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
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 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 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 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. \(( \sqrt { } 6 ) \cos \frac { 1 } { 2 } \theta + ( \sqrt { } 10 ) \sin \frac { 1 } { 2 } \theta = 3\).
CAIE P3 2011 November Q6
8 marks Moderate -0.3
6
  1. Express \(\cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), 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 \(\cos 2 \theta + 3 \sin 2 \theta = 2\), for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2011 November Q3
7 marks Standard +0.3
3
  1. Express \(8 \cos \theta + 15 \sin \theta\) 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 solve the equation \(8 \cos \theta + 15 \sin \theta = 12\), giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2012 November Q2
5 marks Standard +0.3
2
  1. Express \(24 \sin \theta - 7 \cos \theta\) 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.
  2. Hence find the smallest positive value of \(\theta\) satisfying the equation $$24 \sin \theta - 7 \cos \theta = 17$$
CAIE P3 2019 November Q4
7 marks Moderate -0.3
4
  1. Express \(( \sqrt { } 6 ) \sin x + \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 3 decimal places.
  2. Hence solve the equation \(( \sqrt { } 6 ) \sin 2 \theta + \cos 2 \theta = 2\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2019 June Q7
11 marks Standard +0.8
7
    1. Express \(4 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
    2. Hence find the smallest positive value of \(\theta\) satisfying the equation \(4 \sin \theta + 4 \cos \theta = 5\).
  2. Solve the equation $$4 \cot 2 x = 5 + \tan x$$ for \(0 < x < \pi\), showing all necessary working and giving the answers correct to 2 decimal places.
    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 2005 November Q3
7 marks Standard +0.3
3
  1. Express \(12 \cos \theta - 5 \sin \theta\) 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 $$12 \cos \theta - 5 \sin \theta = 10$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2007 November Q6
7 marks Standard +0.3
6
  1. Express \(8 \sin \theta - 15 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), 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 $$8 \sin \theta - 15 \cos \theta = 14$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2009 November Q6
7 marks Moderate -0.3
6
  1. Express \(3 \cos x + 4 \sin x\) in the form \(R \cos ( x - \alpha )\), 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 $$3 \cos x + 4 \sin x = 4.5$$ giving all solutions in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2010 November Q6
7 marks Moderate -0.3
6
  1. Express \(2 \sin \theta - \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), 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 $$2 \sin \theta - \cos \theta = - 0.4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).