1.05o Trigonometric equations: solve in given intervals

Edexcel C3 2016 June Q8

10 marks Standard +0.8

1. Prove that $$2 \cot 2 x + \tan x \equiv \cot x \quad x \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$
2. Hence, or otherwise, solve, for \(- \pi \leqslant x < \pi\), $$6 \cot 2 x + 3 \tan x = \operatorname { cosec } ^ { 2 } x - 2$$ Give your answers to 3 decimal places.
   (Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{d3ba2776-eedb-48f0-834f-41aa454afba3-14_2258_47_315_37}
   

Edexcel C3 2017 June Q4

9 marks Standard +0.3

1. Write \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 \leqslant \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
2. Show that the equation $$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$ can be rewritten in the form $$5 \cos 2 x - 2 \sin 2 x = c$$ where \(c\) is a positive constant to be determined.
3. Hence or otherwise, solve, for \(0 \leqslant x < \pi\), $$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$ giving your answers to 2 decimal places.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

Edexcel C3 2017 June Q9

9 marks Standard +0.3

1. Prove that $$\sin 2 x - \tan x \equiv \tan x \cos 2 x , \quad x \neq ( 2 n + 1 ) 90 ^ { \circ } , \quad n \in \mathbb { Z }$$
2. Given that \(x \neq 90 ^ { \circ }\) and \(x \neq 270 ^ { \circ }\), solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\sin 2 x - \tan x = 3 \tan x \sin x$$ Give your answers in degrees to one decimal place where appropriate.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)
   \includegraphics[max width=\textwidth, alt={}]{f0a633e3-5c63-4d21-8ffa-d4e7dc43a536-32_2632_1826_121_121}

Edexcel C3 2018 June Q6

11 marks Standard +0.3

1. Using the identity for \(\tan ( A \pm B )\), solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), $$\frac { \tan 2 x + \tan 32 ^ { \circ } } { 1 - \tan 2 x \tan 32 ^ { \circ } } = 5$$ Give your answers, in degrees, to 2 decimal places.
2. (a) Using the identity for \(\tan ( A \pm B )\), show that $$\tan \left( 3 \theta - 45 ^ { \circ } \right) \equiv \frac { \tan 3 \theta - 1 } { 1 + \tan 3 \theta } , \quad \theta \neq ( 60 n + 45 ) ^ { \circ } , n \in \mathbb { Z }$$

(b) Hence solve, for \(0 < \theta < 180 ^ { \circ }\), $$( 1 + \tan 3 \theta ) \tan \left( \theta + 28 ^ { \circ } \right) = \tan 3 \theta - 1$$

Edexcel C3 2018 June Q9

9 marks Standard +0.3

1. Express \(\sin \theta - 2 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and the value of \(\alpha\), in radians, to 3 decimal places. $$\mathrm { M } ( \theta ) = 40 + ( 3 \sin \theta - 6 \cos \theta ) ^ { 2 }$$
2. Find
   1. the maximum value of \(\mathrm { M } ( \theta )\),
   2. the smallest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { M } ( \theta )\) occurs. $$N ( \theta ) = \frac { 30 } { 5 + 2 ( \sin 2 \theta - 2 \cos 2 \theta ) ^ { 2 } }$$
   (c) Find
   1. the maximum value of \(\mathrm { N } ( \theta )\),
   2. the largest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { N } ( \theta )\) occurs.
      (Solutions based entirely on graphical or numerical methods are not acceptable.)

      END

Edexcel C3 Q5

11 marks Moderate -0.5

5. (a) Using the formulae $$\begin{gathered} \sin ( A \pm B ) = \sin A \cos B \pm \cos A \sin B \\ \cos ( A \pm B ) = \cos A \cos B \mp \sin A \sin B \end{gathered}$$ show that

1. \(\sin ( A + B ) - \sin ( A - B ) = 2 \cos A \sin B\),
2. \(\cos ( A - B ) - \cos ( A + B ) = 2 \sin A \sin B\).
   (b) Use the above results to show that $$\frac { \sin ( A + B ) - \sin ( A - B ) } { \cos ( A - B ) - \cos ( A + B ) } = \cot A$$ Using the result of part (b) and the exact values of \(\sin 60 ^ { \circ }\) and \(\cos 60 ^ { \circ }\),
   (c) find an exact value for \(\cot 75 ^ { \circ }\) in its simplest form.
   

   5. continued

   Leave blank

Edexcel C3 Q8

12 marks Standard +0.3

1. In a particular circuit the current, \(I\) amperes, is given by

$$I = 4 \sin \theta - 3 \cos \theta , \quad \theta > 0$$ where \(\theta\) is an angle related to the voltage. Given that \(I = R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 \leqslant \alpha < 360 ^ { \circ }\),

1. find the value of \(R\), and the value of \(\alpha\) to 1 decimal place.
2. Hence solve the equation \(4 \sin \theta - 3 \cos \theta = 3\) to find the values of \(\theta\) between 0 and \(360 ^ { \circ }\).
3. Write down the greatest value for \(I\).
4. Find the value of \(\theta\) between 0 and \(360 ^ { \circ }\) at which the greatest value of \(I\) occurs.
   8. continued

Edexcel C3 Specimen Q3

10 marks Standard +0.3

3.

1. Using the identity for \(\cos ( A + B )\), prove that \(\cos \theta \equiv 1 - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).
2. Prove that \(1 + \sin \theta - \cos \theta \equiv 2 \sin \left( \frac { 1 } { 2 } \theta \right) \left[ \cos \left( \frac { 1 } { 2 } \theta \right) + \sin \left( \frac { 1 } { 2 } \theta \right) \right]\).
3. Hence, or otherwise, solve the equation $$1 + \sin \theta - \cos \theta = 0 , \quad 0 \leq \theta < 2 \pi$$

Edexcel F2 2021 January Q8

16 marks Challenging +1.2

8. Given that \(z = e ^ { \mathrm { i } \theta }\)

1. show that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\) where \(n\) is a positive integer.
2. Show that $$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
3. Hence solve the equation $$\cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta = 0 \quad 0 \leqslant \theta \leqslant \pi$$ Give your answers to 3 significant figures.
4. Use calculus to determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \left( 32 \cos ^ { 6 } \theta - 4 \cos ^ { 2 } \theta \right) d \theta$$ Solutions relying entirely on calculator technology are not acceptable.

Edexcel F2 2023 January Q7

8 marks Challenging +1.8

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Use de Moivre's theorem to show that $$\cos 5 x \equiv \cos x \left( a \sin ^ { 4 } x + b \sin ^ { 2 } x + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\cos 5 \theta = \sin 2 \theta \sin \theta - \cos \theta$$ giving your answers to 3 decimal places.

Edexcel FP2 2005 June Q10

12 marks Challenging +1.2

10.

1. Given that \(z = e ^ { \mathrm { i } \theta }\), show that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ where \(n\) is a positive integer.
2. Show that $$\sin ^ { 5 } \theta = \frac { 1 } { 16 } ( \sin 5 \theta - 5 \sin 3 \theta + 10 \sin \theta )$$
3. Hence solve, in the interval \(0 \leq \theta < 2 \pi\), $$\sin 5 \theta - 5 \sin 3 \theta + 6 \sin \theta = 0$$ (5)(Total 12 marks)
   

Edexcel FP2 2006 June Q6

11 marks Challenging +1.2

6.

1. Use de Moivre's theorem to show that \(\boldsymbol { \operatorname { s i n } } 5 \boldsymbol { \theta } = \boldsymbol { \operatorname { s i n } } \boldsymbol { \theta } \left( \mathbf { 1 6 } \mathbf { c o s } ^ { 4 } \boldsymbol { \theta } - \mathbf { 1 2 } \boldsymbol { \operatorname { c o s } } ^ { 2 } \boldsymbol { \theta } + \mathbf { 1 } \right)\).
2. Hence, or otherwise, solve, for \(0 \leq \theta < \pi\) $$\sin 5 \theta + \cos \theta \sin 2 \theta = 0$$ (6)(Total 11 marks)
   

Edexcel FP2 2011 June Q7

11 marks Challenging +1.2

1. Use de Moivre's theorem to show that $$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ Hence, given also that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\),
2. find all the solutions of $$\sin 5 \theta = 5 \sin 3 \theta$$ in the interval \(0 \leqslant \theta < 2 \pi\). Give your answers to 3 decimal places.
   

Edexcel FP2 2013 June Q6

11 marks Standard +0.8

6. The complex number \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta\) is real.

1. Use de Moivre's theorem to show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$ where \(n\) is a positive integer.
2. Show that $$\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
3. Hence find all the solutions of $$\cos 5 \theta + 5 \cos 3 \theta + 12 \cos \theta = 0$$ in the interval \(0 \leqslant \theta < 2 \pi\)

Edexcel FP2 2014 June Q4

10 marks Standard +0.8

4.

1. Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
2. Hence solve for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\) $$64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 3 = 0$$ giving your answers as exact multiples of \(\pi\).
   

Edexcel C3 2012 June Q8

12 marks Standard +0.3

$$f ( x ) = 7 \cos 2 x - 24 \sin 2 x$$ Given that \(\mathrm { f } ( x ) = R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),

1. find the value of \(R\) and the value of \(\alpha\).
2. Hence solve the equation $$7 \cos 2 x - 24 \sin 2 x = 12.5$$ for \(0 \leqslant x < 180 ^ { \circ }\), giving your answers to 1 decimal place.
3. Express \(14 \cos ^ { 2 } x - 48 \sin x \cos x\) in the form \(a \cos 2 x + b \sin 2 x + c\), where \(a , b\), and \(c\) are constants to be found.
4. Hence, using your answers to parts (a) and (c), deduce the maximum value of $$14 \cos ^ { 2 } x - 48 \sin x \cos x$$

OCR C2 2005 January Q5

8 marks Moderate -0.3

5

1. Prove that the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ can be expressed in the form $$2 \cos ^ { 2 } \theta + \cos \theta - 1 = 0$$
2. Hence solve the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

OCR C2 2008 January Q9

9 marks Moderate -0.8

9

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_376_764_276_733} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows the curve \(y = 2 \sin x\) for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). State the coordinates of the maximum and minimum points on this part of the curve.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_371_766_959_731} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Fig. 2 shows the curve \(y = 2 \sin x\) and the line \(y = k\). The smallest positive solution of the equation \(2 \sin x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\), and in the range \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\),
   1. another solution of the equation \(2 \sin x = k\),
   2. one solution of the equation \(2 \sin x = - k\).
   3. Find the \(x\)-coordinates of the points where the curve \(y = 2 \sin x\) intersects the curve \(y = 2 - 3 \cos ^ { 2 } x\), for values of \(x\) such that \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

OCR C2 2005 June Q9

12 marks Standard +0.3

9

1. 1. Write down the exact values of \(\cos \frac { 1 } { 6 } \pi\) and \(\tan \frac { 1 } { 3 } \pi\) (where the angles are in radians). Hence verify that \(x = \frac { 1 } { 6 } \pi\) is a solution of the equation $$2 \cos x = \tan 2 x$$
   2. Sketch, on a single diagram, the graphs of \(y = 2 \cos x\) and \(y = \tan 2 x\), for \(x\) (radians) such that \(0 \leqslant x \leqslant \pi\). Hence state, in terms of \(\pi\), the other values of \(x\) between 0 and \(\pi\) satisfying the equation $$2 \cos x = \tan 2 x$$
2. 1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve \(y = \tan x\), the \(x\)-axis, and the lines \(x = 0.1\) and \(x = 0.4\). (Values of \(x\) are in radians.)
   2. State with a reason whether this approximation is an underestimate or an overestimate.

OCR C2 2006 June Q5

8 marks Moderate -0.3

5 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(2 \sin ^ { 2 } x = 1 + \cos x\).
2. \(\sin 2 x = - \cos 2 x\).

OCR MEI C2 2005 January Q3

4 marks Moderate -0.8

3 Sketch the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.2\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

OCR MEI C2 2006 January Q5

5 marks Moderate -0.8

5

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

OCR MEI C2 2007 January Q6

4 marks Easy -1.2

6 Sketch the curve \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.68\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

OCR MEI C2 2008 January Q6

5 marks Easy -1.2

6

1. Sketch the graph of \(y = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\). Give your answers in the form \(k \pi\).

OCR MEI C2 2007 June Q8

5 marks Moderate -0.8

8

1. Show that the equation \(2 \cos ^ { 2 } \theta + 7 \sin \theta = 5\) may be written in the form $$2 \sin ^ { 2 } \theta - 7 \sin \theta + 3 = 0$$
2. By factorising this quadratic equation, solve the equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\). Section B (36 marks)