1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs

7

1. Show that the equation \(\sqrt { 5 } \sec x + \tan x = 4\) can be expressed as \(R \cos ( x + \alpha ) = \sqrt { 5 }\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [4]
2. Hence solve the equation \(\sqrt { 5 } \sec 2 x + \tan 2 x = 4\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

7

1. Use de Moivre's theorem to show that $$\operatorname { cosec } 7 \theta = \frac { \operatorname { cosec } ^ { 7 } \theta } { 7 \operatorname { cosec } ^ { 6 } \theta - 56 \operatorname { cosec } ^ { 4 } \theta + 112 \operatorname { cosec } ^ { 2 } \theta - 64 }$$
2. Hence obtain the roots of the equation $$x ^ { 7 } - 14 x ^ { 6 } + 112 x ^ { 4 } - 224 x ^ { 2 } + 128 = 0$$ in the form \(\operatorname { cosec } q \pi\), where \(q\) is rational.

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. (a) 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.
(b) 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.

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 $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \right) \cot x d x$$

7. (a) 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,
(b) the minimum height of the seabird above sea level, giving your answer to the nearest cm,
(c) 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}

1. 1. The curve \(C\) has equation \(y = \mathrm { g } ( x )\) where

$$g ( x ) = e ^ { 3 x } \sec 2 x \quad - \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$$

1. Find \(\mathrm { g } ^ { \prime } ( x )\)
2. Hence find the \(x\) coordinate of the stationary point of \(C\).
   (ii) A different curve has equation $$x = \ln ( \sin y ) \quad 0 < y < \frac { \pi } { 2 }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { x } } { \mathrm { f } ( x ) }$$ where \(\mathrm { f } ( x )\) is a function of \(\mathrm { e } ^ { x }\) that should be found.
   

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1. (a) By writing \(\sec x\) as \(\frac { 1 } { \cos x }\), show that \(\frac { \mathrm { d } ( \sec x ) } { \mathrm { d } x } = \sec x \tan x\).

Given that \(y = \mathrm { e } ^ { 2 x } \sec 3 x\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 2 x } \sec 3 x , - \frac { \pi } { 6 } < x < \frac { \pi } { 6 }\), has a minimum turning point at \(( a , b )\).
(c) Find the values of the constants \(a\) and \(b\), giving your answers to 3 significant figures.

8. Solve $$\operatorname { cosec } ^ { 2 } 2 x - \cot 2 x = 1$$ for \(0 \leqslant x \leqslant 180 ^ { \circ }\).

8. (a) Given that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \cos x ) = - \sin x$$ show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \sec x \tan x\). Given that $$x = \sec 2 y$$ (b) find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
(c) Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). \includegraphics[max width=\textwidth, alt={}, center]{3ff6824f-9fbf-4b5b-8bab-91332c549b36-14_102_93_2473_1804}

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.

1. $$f ( x ) = \sec x + 3 x - 2 , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$

1. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \([ 0.2,0.4 ]\)
2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = \frac { 2 } { 3 } - \frac { 1 } { 3 \cos x }$$ The solution of \(\mathrm { f } ( x ) = 0\) is \(\alpha\), where \(\alpha = 0.3\) to 1 decimal place.
3. Starting with \(x _ { 0 } = 0.3\), use the iterative formula $$x _ { n + 1 } = \frac { 2 } { 3 } - \frac { 1 } { 3 \cos x _ { n } }$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
4. State the value of \(\alpha\) correct to 3 decimal places.

1. (a) Prove that

$$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 2 \operatorname { cosec } 2 \theta , \quad \theta \neq 90 n ^ { \circ }$$ (b) On the axes on page 20, sketch the graph of \(y = 2 \operatorname { cosec } 2 \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
(c) Solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 3 ,$$ giving your answers to 1 decimal place. \includegraphics[max width=\textwidth, alt={}, center]{f3c3c777-7808-4d82-a1f4-2dee6674be1e-11_899_1253_315_347}

5. (a) Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \cot ^ { 2 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
(b) Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$2 \cot ^ { 2 } \theta - 9 \operatorname { cosec } \theta = 3$$ giving your answers to 1 decimal place.

2. (a) Use the identity \(\cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 1\) to prove that \(\tan ^ { 2 } \theta = \sec ^ { 2 } \theta - 1\).
(b) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta + 4 \sec \theta + \sec ^ { 2 } \theta = 2$$

8. (a) Write down \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
(b) Find, for \(0 < x < \pi\), all the solutions of the equation $$\operatorname { cosec } x - 8 \cos x = 0$$ giving your answers to 2 decimal places.

3. \(\mathrm { f } ( x ) = 4 \operatorname { cosec } x - 4 x + 1\), where \(x\) is in radians.

1. Show that there is a root \(\alpha\) of \(\mathrm { f } ( x ) = 0\) in the interval [1.2,1.3].
2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = \frac { 1 } { \sin x } + \frac { 1 } { 4 }$$
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { \sin x _ { n } } + \frac { 1 } { 4 } , \quad x _ { 0 } = 1.25$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
4. By considering the change of sign of \(\mathrm { f } ( x )\) in a suitable interval, verify that \(\alpha = 1.291\) correct to 3 decimal places.

6. (a) Prove that $$\frac { 1 } { \sin 2 \theta } - \frac { \cos 2 \theta } { \sin 2 \theta } = \tan \theta , \quad \theta \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise,

1. show that \(\tan 15 ^ { \circ } = 2 - \sqrt { 3 }\),
2. solve, for \(0 < x < 360 ^ { \circ }\), $$\operatorname { cosec } 4 x - \cot 4 x = 1$$

1. (a) Express \(4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
   (b) Hence show that

$$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = \sec ^ { 2 } \theta$$ (c) Hence or otherwise solve, for \(0 < \theta < \pi\), $$4 \operatorname { cosec } ^ { 2 } 2 \theta - \operatorname { cosec } ^ { 2 } \theta = 4$$ giving your answers in terms of \(\pi\).

1. (i) Use an appropriate double angle formula to show that

$$\operatorname { cosec } 2 x = \lambda \operatorname { cosec } x \sec x$$ and state the value of the constant \(\lambda\).
(ii) Solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$3 \sec ^ { 2 } \theta + 3 \sec \theta = 2 \tan ^ { 2 } \theta$$ You must show all your working. Give your answers in terms of \(\pi\).

2. Given that \(\tan 40 ^ { \circ } = p\), find in terms of \(p\)

1. \(\cot 40 ^ { \circ }\)
2. \(\sec 40 ^ { \circ }\)
3. \(\tan 85 ^ { \circ }\)