1.05o Trigonometric equations: solve in given intervals

1022 questions

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Edexcel C34 2017 January Q11
11 marks Standard +0.3
  1. Express \(35 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures.
  2. Hence solve, for \(0 \leqslant x < 2 \pi\), $$70 \sin x - 24 \cos x = 37$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) $$y = \frac { 7000 } { 31 + ( 35 \sin x - 12 \cos x ) ^ { 2 } } , \quad x > 0$$
  3. Use your answer to part (a) to calculate
    1. the minimum value of \(y\),
    2. the smallest value of \(x , x > 0\), at which this minimum value occurs.
Edexcel C34 2018 January Q9
9 marks Standard +0.3
  1. Show that $$\frac { \cot ^ { 2 } x } { 1 + \cot ^ { 2 } x } \equiv \cos ^ { 2 } x$$
  2. Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\frac { \cot ^ { 2 } x } { 1 + \cot ^ { 2 } x } = 8 \cos 2 x + 2 \cos x$$ Give each solution in degrees to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2018 January Q12
12 marks Standard +0.3
  1. Express \(2 \sin x - 4 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 significant figures. In a town in Norway, a student records the number of hours of daylight every day for a year. He models the number of hours of daylight, \(H\), by the continuous function given by the formula $$H = 12 + 4 \sin \left( \frac { 2 \pi t } { 365 } \right) - 8 \cos \left( \frac { 2 \pi t } { 365 } \right) , \quad 0 \leqslant t \leqslant 365$$ where \(t\) is the number of days since he began recording.
  2. Using your answer to part (a), or otherwise, find the maximum and minimum number of hours of daylight given by this formula. Give your answers to 3 significant figures.
  3. Use the formula to find the values of \(t\) when \(H = 17\), giving your answers to the nearest integer.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
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Edexcel C34 2019 January Q1
12 marks Standard +0.3
  1. Express \(7 \sin 2 \theta - 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
  2. Hence solve, for \(0 \leqslant \theta < 90 ^ { \circ }\), the equation $$7 \sin 2 \theta - 2 \cos 2 \theta = 4$$ giving your answers in degrees to one decimal place.
  3. Express \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\) in the form \(a \sin 2 \theta + b \cos 2 \theta + c\), where \(a\), \(b\) and \(c\) are constants to be found.
  4. Use your answers to part (a) and part (c) to deduce the exact maximum value of \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\)
Edexcel C34 2015 June Q7
8 marks Standard +0.8
7.
  1. Use the substitution \(t = \tan x\) to show that the equation $$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$ can be written in the form $$3 t ^ { 4 } + 8 t ^ { 2 } - 3 = 0$$
  2. Hence solve, for \(0 \leqslant x < 2 \pi\), $$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$ Give each answer in terms of \(\pi\). You must make your method clear.
Edexcel C34 2017 June Q7
9 marks Standard +0.3
  1. Prove that $$\frac { 1 - \cos 2 x } { 1 + \cos 2 x } \equiv \tan ^ { 2 } x , \quad x \neq ( 2 n + 1 ) 90 ^ { \circ } , n \in \mathbb { Z }$$
  2. Hence, or otherwise, solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), $$\frac { 2 - 2 \cos 2 \theta } { 1 + \cos 2 \theta } - 2 = 7 \sec \theta$$ Give your answers in degrees to one decimal place.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2018 June Q6
11 marks Standard +0.3
6.
  1. Express \(\sqrt { 5 } \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 significant figures.
  2. Solve, for \(- \pi < \theta < \pi\), $$\sqrt { 5 } \cos \theta - 2 \sin \theta = 0.5$$ giving your answers to 3 significant figures. [Solutions based entirely on graphical or numerical methods are not acceptable.] $$\mathrm { f } ( x ) = A ( \sqrt { 5 } \cos \theta - 2 \sin \theta ) + B \quad \theta \in \mathbb { R }$$ where \(A\) and \(B\) are constants. Given that the range of f is $$- 15 \leqslant f ( x ) \leqslant 33$$
  3. find the value of \(B\) and the possible values of \(A\).
Edexcel C34 2018 June Q12
9 marks Standard +0.8
12.
  1. Show that $$\cot x - \tan x \equiv 2 \cot 2 x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$
  2. Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$5 + \cot \left( \theta - 15 ^ { \circ } \right) - \tan \left( \theta - 15 ^ { \circ } \right) = 0$$ giving your answers to one decimal place.
    [0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C34 2019 June Q7
9 marks Standard +0.3
7.
  1. Express \(5 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 4 decimal places. The height of sea water, \(H\) metres, on a harbour wall is modelled by the equation $$H = 6 + 2.5 \cos \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \sin \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) is the number of hours after midday.
  2. Calculate the times at which the model predicts that the height of sea water on the harbour wall will be 4.6 metres. Give your answers to the nearest minute. \includegraphics[max width=\textwidth, alt={}, center]{a9870c94-0910-46ec-a54a-44a431cb324e-18_2257_54_314_1977}
Edexcel C34 2017 October Q4
9 marks Standard +0.3
4.
  1. Prove that $$\frac { 1 - \cos 2 x } { \sin 2 x } \equiv \tan x , \quad x \neq \frac { n \pi } { 2 }$$
  2. Hence solve, for \(0 \leqslant \theta < 2 \pi\), $$3 \sec ^ { 2 } \theta - 7 = \frac { 1 - \cos 2 \theta } { \sin 2 \theta }$$ Give your answers in radians to 3 decimal places, as appropriate.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2018 October Q1
8 marks Standard +0.3
  1. Write \(\cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
  2. Hence solve, for \(0 \leqslant \theta < \pi\), the equation $$\cos 2 \theta + 4 \sin 2 \theta = 1.2$$ giving your answers to 2 decimal places.
Edexcel C34 Specimen Q1
8 marks Standard +0.3
  1. Express \(5 \cos 2 \theta - 12 \sin 2 \theta\) in the form \(R \cos ( 2 \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\) Give the value of \(\alpha\) to 2 decimal places.
  2. Hence solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$5 \cos 2 \theta - 12 \sin 2 \theta = 10$$ giving your answers to 1 decimal place.
Edexcel C34 Specimen Q7
10 marks Challenging +1.2
  1. Show that $$\cot x - \cot 2 x \equiv \operatorname { cosec } 2 x , \quad x \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$
  2. Hence, or otherwise, solve for \(0 \leqslant \theta \leqslant \pi\) $$\operatorname { cosec } \left( 3 \theta + \frac { \pi } { 3 } \right) + \cot \left( 3 \theta + \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 3 } }$$ You must show your working.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C3 2006 January Q6
12 marks Standard +0.3
6. $$f ( x ) = 12 \cos x - 4 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x + \alpha )\), where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\),
  1. find the value of \(R\) and the value of \(\alpha\).
  2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
    1. Write down the minimum value of \(12 \cos x - 4 \sin x\).
    2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs. \includegraphics[max width=\textwidth, alt={}, center]{5cd53af1-bac9-4ed9-ac45-59ad2e372423-09_60_35_2669_1853}
Edexcel C3 2006 January Q7
12 marks Standard +0.3
7. (a) Show that
  1. \(\frac { \cos 2 x } { \cos x + \sin x } \equiv \cos x - \sin x , \quad x \neq \left( n - \frac { 1 } { 4 } \right) \pi , n \in \mathbb { Z }\),
  2. \(\frac { 1 } { 2 } ( \cos 2 x - \sin 2 x ) \equiv \cos ^ { 2 } x - \cos x \sin x - \frac { 1 } { 2 }\).
    (b) Hence, or otherwise, show that the equation $$\cos \theta \left( \frac { \cos 2 \theta } { \cos \theta + \sin \theta } \right) = \frac { 1 } { 2 }$$ can be written as $$\sin 2 \theta = \cos 2 \theta$$ (c) Solve, for \(0 \leqslant \theta < 2 \pi\), $$\sin 2 \theta = \cos 2 \theta$$ giving your answers in terms of \(\pi\).
Edexcel C3 2007 January Q5
8 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{a4ad749b-181b-4680-8771-94d9b581125a-07_865_926_301_516}
\end{figure} Figure 1 shows an oscilloscope screen. The curve shown on the screen satisfies the equation $$y = \sqrt { 3 } \cos x + \sin x$$
  1. Express the equation of the curve in the form \(y = R \sin ( x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
  2. Find the values of \(x , 0 \leqslant x < 2 \pi\), for which \(y = 1\).
Edexcel C3 2008 January Q6
11 marks Standard +0.3
6.
  1. Use the double angle formulae and the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to obtain an expression for \(\cos 3 x\) in terms of powers of \(\cos x\) only.
    1. Prove that $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } \equiv 2 \sec x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$
    2. Hence find, for \(0 < x < 2 \pi\), all the solutions of $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } = 4$$
Edexcel C3 2008 January Q7
13 marks Standard +0.3
  1. A curve \(C\) has equation
$$y = 3 \sin 2 x + 4 \cos 2 x , - \pi \leqslant x \leqslant \pi$$ The point \(A ( 0,4 )\) lies on \(C\).
  1. Find an equation of the normal to the curve \(C\) at \(A\).
  2. Express \(y\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 significant figures.
  3. Find the coordinates of the points of intersection of the curve \(C\) with the \(x\)-axis. Give your answers to 2 decimal places.
Edexcel C3 2009 January Q6
13 marks Standard +0.3
6.
    1. By writing \(3 \theta = ( 2 \theta + \theta )\), show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
    2. Hence, or otherwise, for \(0 < \theta < \frac { \pi } { 3 }\), solve $$8 \sin ^ { 3 } \theta - 6 \sin \theta + 1 = 0 .$$ Give your answers in terms of \(\pi\).
  2. Using \(\sin ( \theta - \alpha ) = \sin \theta \cos \alpha - \cos \theta \sin \alpha\), or otherwise, show that $$\sin 15 ^ { \circ } = \frac { 1 } { 4 } ( \sqrt { } 6 - \sqrt { } 2 )$$
Edexcel C3 2009 January Q8
12 marks Standard +0.3
8.
  1. Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
  2. Hence find the maximum value of \(3 \cos \theta + 4 \sin \theta\) and the smallest positive value of \(\theta\) for which this maximum occurs. The temperature, \(\mathrm { f } ( t )\), of a warehouse is modelled using the equation $$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$ where \(t\) is the time in hours from midday and \(0 \leqslant t < 24\).
  3. Calculate the minimum temperature of the warehouse as given by this model.
  4. Find the value of \(t\) when this minimum temperature occurs.
Edexcel C3 2010 January Q8
7 marks Standard +0.3
8. Solve $$\operatorname { cosec } ^ { 2 } 2 x - \cot 2 x = 1$$ for \(0 \leqslant x \leqslant 180 ^ { \circ }\).
Edexcel C3 2011 January Q3
6 marks Moderate -0.3
  1. Find all the solutions of
$$2 \cos 2 \theta = 1 - 2 \sin \theta$$ in the interval \(0 \leqslant \theta < 360 ^ { \circ }\).
Edexcel C3 2012 January Q5
10 marks Standard +0.3
5. Solve, for \(0 \leqslant \theta \leqslant 180 ^ { \circ }\), $$2 \cot ^ { 2 } 3 \theta = 7 \operatorname { cosec } 3 \theta - 5$$ Give your answers in degrees to 1 decimal place.
Edexcel C3 2012 January Q8
13 marks Standard +0.3
8.
  1. Starting from the formulae for \(\sin ( A + B )\) and \(\cos ( A + B )\), prove that
  2. Deduce that $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$
  3. Hence, or otherwise, solve, for \(0 \leqslant \theta \leqslant \pi\), $$\tan \left( \theta + \frac { \pi } { 6 } \right) = \frac { 1 + \sqrt { } 3 \tan \theta } { \sqrt { } 3 - \tan \theta }$$
(c) Hence, or otherwise, solve, for \(0 \leqslant \theta \leqslant \pi\),
(c) $$1 + \sqrt { } 3 \tan \theta = ( \sqrt { } 3 - \tan \theta ) \tan ( \pi - \theta )$$ \section*{}
Edexcel C3 2013 January Q6
11 marks Standard +0.3
6.
  1. Without using a calculator, find the exact value of $$\left( \sin 22.5 ^ { \circ } + \cos 22.5 ^ { \circ } \right) ^ { 2 }$$ You must show each stage of your working.
  2. (a) Show that \(\cos 2 \theta + \sin \theta = 1\) may be written in the form $$k \sin ^ { 2 } \theta - \sin \theta = 0 , \text { stating the value of } k$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$\cos 2 \theta + \sin \theta = 1$$