Solve equation with reciprocal functions

2

1. Express the equation \(7 \tan \theta + 4 \cot \theta - 13 \sec \theta = 0\) in terms of \(\sin \theta\) only.
2. Hence solve the equation \(7 \tan \theta + 4 \cot \theta - 13 \sec \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

1 Solve the equation $$\sec ^ { 2 } \theta + 5 \tan ^ { 2 } \theta = 9 + 17 \sec \theta$$ for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

2 Solve the equation \(\sec ^ { 2 } \theta \cot \theta = 8\) for \(0 < \theta < \pi\).

1 Solve the equation \(7 \cot \theta = 3 \operatorname { cosec } \theta\) for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

1 Solve the equation \(\sec \theta = 5 \operatorname { cosec } \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

5 Solve the equation \(\sec x = 4 - 2 \tan ^ { 2 } x\), giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

3 By expressing the equation \(\operatorname { cosec } \theta = 3 \sin \theta + \cot \theta\) in terms of \(\cos \theta\) only, solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

1 Solve the equation \(\sec ^ { 2 } \theta + \tan ^ { 2 } \theta = 5 \tan \theta + 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\). Show all necessary working.

5 Solve the equation \(8 + \cot \theta = 2 \operatorname { cosec } ^ { 2 } \theta\), giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3 Solve the equation \(2 \cot ^ { 2 } \theta - 5 \operatorname { cosec } \theta = 10\), giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3 Solve the equation \(\sec ^ { 2 } \theta = 3 \operatorname { cosec } \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

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 $$\frac { \sin 3 \theta } { \sin \theta } - \frac { \cos 3 \theta } { \cos \theta } \equiv 2$$

   VIAV SIHI NI III IM I ON OC

   VIIIV SIHI NI JIIIM I ON OO

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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. (a) Show that

$$\cot ^ { 2 } x - \operatorname { cosec } x - 11 = 0$$ may be expressed in the form \(\operatorname { cosec } ^ { 2 } x - \operatorname { cosec } x + k = 0\), where \(k\) is a constant.
(b) Hence solve for \(0 \leqslant x < 360 ^ { \circ }\) $$\cot ^ { 2 } x - \operatorname { cosec } x - 11 = 0$$ Give each solution in degrees to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

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

1. (a) Using \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(\operatorname { cosec } ^ { 2 } \theta - \cot ^ { 2 } \theta \equiv 1\).
   (b) Hence, or otherwise, prove that

$$\operatorname { cosec } ^ { 4 } \theta - \cot ^ { 4 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta + \cot ^ { 2 } \theta$$ (c) Solve, for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\), $$\operatorname { cosec } ^ { 4 } \theta - \cot ^ { 4 } \theta = 2 - \cot \theta$$

1. Find, to 2 decimal places, the solutions of the equation

$$3 \cot ^ { 2 } x - 4 \operatorname { cosec } x + \operatorname { cosec } ^ { 2 } x = 0$$ in the interval \(0 \leq x \leq 2 \pi\).

3. Solve, for \(0 \leq y \leq 360\), the equation $$2 \cot ^ { 2 } y ^ { \circ } + 5 \operatorname { cosec } y ^ { \circ } + \operatorname { cosec } ^ { 2 } y ^ { \circ } = 0$$

2. Solve the equation $$3 \operatorname { cosec } \theta ^ { \circ } + 8 \cos \theta ^ { \circ } = 0$$ for \(\theta\) in the interval \(0 \leq \theta \leq 180\), giving your answers to 1 decimal place.

6 Solve the equation \(\operatorname { cosec } \theta = 3\), for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

8 Solve the equation $$\sec ^ { 2 } \theta = 4 , \quad 0 \leqslant \theta \leqslant \pi ,$$ giving your answers in terms of \(\pi\).

3 Solve the equation $$\sec ^ { 2 } \theta = 4 , \quad 0 \leqslant \theta \leqslant \pi ,$$ giving your answers in terms of \(\pi\).

9

1. It is given that \(\sec x - \tan x = - 5\).
   1. Show that \(\sec x + \tan x = - 0.2\).
   2. Hence find the exact value of \(\cos x\).
2. Hence solve the equation $$\sec \left( 2 x - 70 ^ { \circ } \right) - \tan \left( 2 x - 70 ^ { \circ } \right) = - 5$$ giving all values of \(x\), to one decimal place, in the interval \(- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
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