1.05o Trigonometric equations: solve in given intervals

5.

1. Use the substitution \(t = \tan x\) to show that the equation $$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$ can be written in the form $$5 t ^ { 4 } - 24 t ^ { 2 } - 5 = 0$$
2. Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$ Show each stage of your working and give your answers to one decimal place.
   

7.

1. Prove that $$\frac { \sin 2 x } { \cos x } + \frac { \cos 2 x } { \sin x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } n \in \mathbb { Z }$$
2. Hence solve, for \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\) $$7 + \frac { \sin 4 \theta } { \cos 2 \theta } + \frac { \cos 4 \theta } { \sin 2 \theta } = 3 \cot ^ { 2 } 2 \theta$$ giving your answers in radians to 3 significant figures where appropriate.
   

2.

1. Show that the equation $$8 \cos \theta = 3 \operatorname { cosec } \theta$$ can be written in the form $$\sin 2 \theta = k$$ where \(k\) is a constant to be found.
2. Hence find the smallest positive solution of the equation $$8 \cos \theta = 3 \operatorname { cosec } \theta$$ giving your answer, in degrees, to one decimal place.

9. In this question you must show detailed reasoning. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 < x \leqslant \pi\), the equation $$2 \sec ^ { 2 } x - 3 \tan x = 2$$ giving the answers, as appropriate, to 3 significant figures.
2. Prove that $$\frac { \sin 3 \theta } { \sin \theta } - \frac { \cos 3 \theta } { \cos \theta } \equiv 2$$

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

Solutions relying entirely on calculator technology are not acceptable.

1. Prove that $$\cot ^ { 2 } x - \tan ^ { 2 } x \equiv 4 \cot 2 x \operatorname { cosec } 2 x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\) $$4 \cot 2 \theta \operatorname { cosec } 2 \theta = 2 \tan ^ { 2 } \theta$$ giving your answers to 2 decimal places.

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 $$\frac { 3 \sin \theta \cos \theta } { \cos \theta + \sin \theta } = ( 2 + \sec 2 \theta ) ( \cos \theta - \sin \theta )$$ can be written in the form $$3 \sin 2 \theta - 4 \cos 2 \theta = 2$$
2. Hence solve for \(\pi < x < \frac { 3 \pi } { 2 }\) $$\frac { 3 \sin x \cos x } { \cos x + \sin x } = ( 2 + \sec 2 x ) ( \cos x - \sin x )$$ giving the answer to 3 significant figures.

2.

1. Show that $$\frac { 1 - \cos 2 x } { 2 \sin 2 x } \equiv k \tan x \quad x \neq ( 90 n ) ^ { \circ } \quad n \in \mathbb { Z }$$ where \(k\) is a constant to be found.
2. Hence solve, for \(0 < \theta < 90 ^ { \circ }\) $$\frac { 9 ( 1 - \cos 2 \theta ) } { 2 \sin 2 \theta } = 2 \sec ^ { 2 } \theta$$ giving your answers to one decimal place.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)
   

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

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

1. Show that the equation $$2 \sin \theta \left( 3 \cot ^ { 2 } 2 \theta - 7 \right) = 13 \sec \theta$$ can be written as $$3 \operatorname { cosec } ^ { 2 } 2 \theta - 13 \operatorname { cosec } 2 \theta - 10 = 0$$
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\), the equation $$2 \sin \theta \left( 3 \cot ^ { 2 } 2 \theta - 7 \right) = 13 \sec \theta$$ giving your answers to 3 significant figures.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 < x < \pi\) $$( x - 2 ) ( \sqrt { 3 } \sec x + 2 ) = 0$$
2. Solve, for \(0 < \theta < 360 ^ { \circ }\) $$10 \sin \theta = 3 \cos 2 \theta$$

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\left( \frac { \cos 2 \theta } { \sin \theta } + \frac { \sin 2 \theta } { \cos \theta } \right) ^ { 2 } = 6 \cot \theta - 4$$ giving your answers to 3 significant figures as appropriate.
3. Using the result from part (a), or otherwise, find the exact value of $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \right) \cot x d x$$

4. $$f ( x ) = 8 \sin x \cos x + 4 \cos ^ { 2 } x - 3$$

1. Write \(\mathrm { f } ( x )\) in the form $$a \sin 2 x + b \cos 2 x + c$$ where \(a\), \(b\) and \(c\) are integers to be found.
2. Use the answer to part (a) to write \(\mathrm { f } ( x )\) in the form $$R \sin ( 2 x + \alpha ) + c$$ 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.
3. Hence, or otherwise,
   1. state the maximum value of \(\mathrm { f } ( x )\)
   2. find the second smallest positive value of \(x\) at which a maximum value of \(\mathrm { f } ( x )\) occurs. Give your answer to 3 significant figures.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Given that $$\sqrt { 2 } \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 60 ^ { \circ } \right)$$ show that $$\tan x = - 2 - \sqrt { 3 }$$
2. Hence or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$\sqrt { 2 } \sin ( 2 \theta ) = \cos \left( 2 \theta - 105 ^ { \circ } \right)$$

1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation

$$2 \cos 2 x = 7 \cos x$$ giving your solutions to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7.

1. Express \(\cos x + 4 \sin x\) in the form \(R \cos ( x - \alpha )\) 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 decimal places. A scientist is studying the behaviour of seabirds in a colony. She models the height above sea level, \(H\) metres, of one of the birds in the colony by the equation $$H = \frac { 24 } { 3 + \cos \left( \frac { 1 } { 2 } t \right) + 4 \sin \left( \frac { 1 } { 2 } t \right) } \quad 0 \leqslant t \leqslant 6.5$$ where \(t\) seconds is the time after it leaves the nest. Find, according to the model,
2. the minimum height of the seabird above sea level, giving your answer to the nearest cm,
3. the value of \(t\), to 2 decimal places, when \(H = 10\) \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-21_2255_50_314_34}
   

4. In this question you should show detailed reasoning. \section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Show that the equation $$2 \sin \left( \theta - 30 ^ { \circ } \right) = 5 \cos \theta$$ can be written in the form $$\tan \theta = 2 \sqrt { 3 }$$
2. Hence, or otherwise, solve for \(0 \leqslant x \leqslant 360 ^ { \circ }\) $$2 \sin \left( x - 10 ^ { \circ } \right) = 5 \cos \left( x + 20 ^ { \circ } \right)$$ giving your answers to one decimal place.

9. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Given that \(\cos 2 \theta - \sin 3 \theta \neq 0\)

1. prove that $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } \equiv \frac { 1 + \sin \theta } { 1 - 2 \sin \theta - 4 \sin ^ { 2 } \theta }$$
2. Hence solve, for \(0 < \theta \leqslant 360 ^ { \circ }\) $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } = 2 \operatorname { cosec } \theta$$ Give your answers to one decimal place. \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-28_2257_52_309_1983}

9.

1. Prove that $$\sec 2 A + \tan 2 A \equiv \frac { \cos A + \sin A } { \cos A - \sin A } \quad A \neq \frac { ( 2 n + 1 ) \pi } { 4 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(0 \leqslant \theta < 2 \pi\) $$\sec 2 \theta + \tan 2 \theta = \frac { 1 } { 2 }$$ Give your answers to 3 decimal places.

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2. Solve, for \(0 \leqslant x \leqslant 270 ^ { \circ }\), the equation $$\frac { \tan 2 x + \tan 50 ^ { \circ } } { 1 - \tan 2 x \tan 50 ^ { \circ } } = 2$$ Give your answers in degrees to 2 decimal places.
(6) \includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-05_104_95_2613_1786}

8.

1. Prove that $$\text { 2cosec } 2 A - \cot A \equiv \tan A , \quad A \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$
2. Hence solve, for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\)
   1. \(2 \operatorname { cosec } 4 \theta - \cot 2 \theta = \sqrt { } 3\)
   2. \(\tan \theta + \cot \theta = 5\) Give your answers to 3 significant figures.

2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$2 \cos 2 \theta = 5 - 13 \sin \theta$$ Give your answers in radians to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7.

1. Given that $$2 \cos ( x + 30 ) ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ without using a calculator, show that $$\tan x ^ { \circ } = 3 \sqrt { 3 } - 4$$
2. Hence or otherwise solve, for \(0 \leqslant \theta < 180\), $$2 \cos ( 2 \theta + 40 ) ^ { \circ } = \sin ( 2 \theta - 20 ) ^ { \circ }$$ Give your answers to one decimal place.
   

10.

1. Express \(3 \sin 2 x + 5 \cos 2 x\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) to 3 significant figures.
2. Solve, for \(0 < x < \pi\), $$3 \sin 2 x + 5 \cos 2 x = 4$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) $$g ( x ) = 4 ( 3 \sin 2 x + 5 \cos 2 x ) ^ { 2 } + 3$$
3. Using your answer to part (a) and showing your working,
   1. find the greatest value of \(\mathrm { g } ( x )\),
   2. find the least value of \(\mathrm { g } ( x )\).

8.

1. Using the trigonometric identity for \(\tan ( A + B )\), prove that $$\tan 3 x = \frac { 3 \tan x - \tan ^ { 3 } x } { 1 - 3 \tan ^ { 2 } x } , \quad x \neq ( 2 n + 1 ) 30 ^ { \circ } , \quad n \in \mathbb { Z }$$
2. Hence solve, for \(- 30 ^ { \circ } < x < 30 ^ { \circ }\), $$\tan 3 x = 11 \tan x$$ (Solutions based entirely on graphical or numerical methods are not acceptable.)