Convert equation to quadratic form

6 Given that \(\operatorname { cosec } ^ { 2 } \theta - \cot \theta = 3\), show that \(\cot ^ { 2 } \theta - \cot \theta - 2 = 0\).
Hence solve the equation \(\operatorname { cosec } ^ { 2 } \theta - \cot \theta = 3\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$2 \tan \theta \left( 8 \cos \theta + 23 \sin ^ { 2 } \theta \right) = 8 \sin 2 \theta \left( 1 + \tan ^ { 2 } \theta \right)$$ may be written as $$\sin 2 \theta \left( A \cos ^ { 2 } \theta + B \cos \theta + C \right) = 0$$ where \(A , B\) and \(C\) are constants to be found.
2. Hence, solve for \(360 ^ { \circ } \leqslant x \leqslant 540 ^ { \circ }\) $$2 \tan x \left( 8 \cos x + 23 \sin ^ { 2 } x \right) = 8 \sin 2 x \left( 1 + \tan ^ { 2 } x \right) \quad x \in \mathbb { R } \quad x \neq 450 ^ { \circ }$$

5

1. Show that the equation \(2 \cos x \tan ^ { 2 } x = 3 ( 1 + \cos x )\) can be expressed in the form $$5 \cos ^ { 2 } x + 3 \cos x - 2 = 0$$ \section*{(b) In this question you must show detailed reasoning.} Hence solve the equation $$2 \cos 3 \theta \tan ^ { 2 } 3 \theta = 3 ( 1 + \cos 3 \theta ) ,$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(120 ^ { \circ }\), correct to \(\mathbf { 1 }\) decimal place where appropriate.

5

1. 1. Show that the equation $$2 \cot ^ { 2 } x + 5 \operatorname { cosec } x = 10$$ can be written in the form \(2 \operatorname { cosec } ^ { 2 } x + 5 \operatorname { cosec } x - 12 = 0\).
   2. Hence show that \(\sin x = - \frac { 1 } { 4 }\) or \(\sin x = \frac { 2 } { 3 }\).
2. Hence, or otherwise, solve the equation $$2 \cot ^ { 2 } ( \theta - 0.1 ) + 5 \operatorname { cosec } ( \theta - 0.1 ) = 10$$ giving all values of \(\theta\) in radians to two decimal places in the interval \(- \pi < \theta < \pi\).
   (3 marks)

2

1. Solve the equation \(\cot x = 2\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.
2. Show that the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\) can be written as $$2 \cot ^ { 2 } x - 3 \cot x - 2 = 0$$
3. Solve the equation \(\operatorname { cosec } ^ { 2 } x = \frac { 3 \cot x + 4 } { 2 }\), giving all values of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) in radians to two decimal places.

7

1. Solve the equation \(\sec x = - 5\), giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
2. Show that the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ can be written in the form $$\sec ^ { 2 } x = 25$$
3. Hence, or otherwise, solve the equation $$\frac { \operatorname { cosec } x } { 1 + \operatorname { cosec } x } - \frac { \operatorname { cosec } x } { 1 - \operatorname { cosec } x } = 50$$ giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < 2 \pi\).
   (3 marks)

6

1. Show that $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) }$$ can be written as \(\operatorname { cosec } ^ { 2 } x\).
2. Hence solve the equation $$\frac { \sec ^ { 2 } x } { ( \sec x + 1 ) ( \sec x - 1 ) } = \operatorname { cosec } x + 3$$ giving the values of \(x\) to the nearest degree in the interval \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).
3. Hence solve the equation $$\frac { \sec ^ { 2 } \left( 2 \theta - 60 ^ { \circ } \right) } { \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) + 1 \right) \left( \sec \left( 2 \theta - 60 ^ { \circ } \right) - 1 \right) } = \operatorname { cosec } \left( 2 \theta - 60 ^ { \circ } \right) + 3$$ giving the values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

4 It is given that \(\tan ^ { 2 } x = \sec x + 11\).

1. Show that the equation \(\tan ^ { 2 } x = \sec x + 11\) can be written in the form $$\sec ^ { 2 } x - \sec x - 12 = 0$$
2. Hence show that \(\cos x = \frac { 1 } { 4 }\) or \(\cos x = - \frac { 1 } { 3 }\).
3. Hence, or otherwise, solve the equation \(\tan ^ { 2 } x = \sec x + 11\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

5

1. Show that the equation $$10 \operatorname { cosec } ^ { 2 } x = 16 - 11 \cot x$$ can be written in the form $$10 \cot ^ { 2 } x + 11 \cot x - 6 = 0$$
2. Hence, given that \(10 \operatorname { cosec } ^ { 2 } x = 16 - 11 \cot x\), find the possible values of \(\tan x\).

8

1. Show that the equation $$\frac { 1 } { 1 + \cos \theta } + \frac { 1 } { 1 - \cos \theta } = 32$$ can be written in the form $$\operatorname { cosec } ^ { 2 } \theta = 16$$
2. Hence, or otherwise, solve the equation $$\frac { 1 } { 1 + \cos ( 2 x - 0.6 ) } + \frac { 1 } { 1 - \cos ( 2 x - 0.6 ) } = 32$$ giving all values of \(x\) in radians to two decimal places in the interval \(0 < x < \pi\).
   (5 marks)

8

1. Show that the equation \(4 \operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta = k\), where \(k \neq 4\), can be written in the form $$\sec ^ { 2 } \theta = \frac { k - 1 } { k - 4 }$$
2. Hence, or otherwise, solve the equation $$4 \operatorname { cosec } ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) - \cot ^ { 2 } \left( 2 x + 75 ^ { \circ } \right) = 5$$ giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
   [0pt] [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-18_72_113_1055_159}
   
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It is given that \(\theta\) is an acute angle measured in degrees such that $$2\sec^2\theta + 3\tan\theta = 22.$$

1. Find the value of \(\tan\theta\). [3]
2. Use an appropriate formula to find the exact value of \(\tan(\theta + 135°)\). [3]