Deduce related solution

7

1. Solve the equation \(2 \cos ^ { 2 } \theta = 3 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
2. The smallest positive solution of the equation \(2 \cos ^ { 2 } ( n \theta ) = 3 \sin ( n \theta )\), where \(n\) is a positive integer, is \(10 ^ { \circ }\). State the value of \(n\) and hence find the largest solution of this equation in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

1. Solve the equation \(4 \sin ^ { 2 } x + 8 \cos x - 7 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Hence find the solution of the equation \(4 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) + 8 \cos \left( \frac { 1 } { 2 } \theta \right) - 7 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

8. (a) Given that \(7 \sin x = 3 \cos x\), find the exact value of \(\tan x\).
(b) Hence solve for \(0 \leqslant \theta < 360 ^ { \circ }\) $$7 \sin \left( 2 \theta + 30 ^ { \circ } \right) = 3 \cos \left( 2 \theta + 30 ^ { \circ } \right)$$ 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.

Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\) the equation $$2 \tan \theta + 3 \sin \theta = 0$$ giving your answers, as appropriate, to one decimal place.
2. Hence, or otherwise, find the smallest positive solution of $$2 \tan \left( 2 x + 40 ^ { \circ } \right) + 3 \sin \left( 2 x + 40 ^ { \circ } \right) = 0$$ giving your answer to one decimal place.

5. (a) Given that \(5 \sin \theta = 2 \cos \theta\), find the value of \(\tan \theta\).
(b) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin 2 x = 2 \cos 2 x$$ giving your answers to 1 decimal place.

3. (i) Given that $$5 \cos \theta - 2 \sin \theta = 0$$ show that \(\tan \theta = 2.5\) (ii) Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2 x ^ { \circ } - 2 \sin 2 x ^ { \circ } = 0 ,$$ giving your answers to 1 decimal place.

14. Given that $$2 \cos ( x + 60 ) ^ { 0 } = \sin ( x - 30 ) ^ { 0 }$$ a. Show, without using a calculator, that $$\tan x = \frac { \sqrt { 3 } } { 3 }$$ b. Hence solve, for \(0 \leq \theta < 360 ^ { 0 }\) $$2 \cos ( 2 \theta + 90 ) ^ { 0 } = \sin ( 2 \theta ) ^ { 0 }$$

1. (a) Solve, for \(- 180 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), the equation

$$5 \sin 2 \theta = 9 \tan \theta$$ giving your answers, where necessary, to one decimal place.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]
(b) Deduce the smallest positive solution to the equation $$5 \sin \left( 2 x - 50 ^ { \circ } \right) = 9 \tan \left( x - 25 ^ { \circ } \right)$$

1. (a) Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), the equation

$$3 \sin ^ { 2 } x + \sin x + 8 = 9 \cos ^ { 2 } x$$ giving your answers to 2 decimal places.
(b) Hence find the smallest positive solution of the equation $$3 \sin ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) + \sin \left( 2 \theta - 30 ^ { \circ } \right) + 8 = 9 \cos ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right)$$ giving your answer to 2 decimal places.

8

1. 1. Show that the equation $$3 \sin \theta \tan \theta = 5 \cos \theta - 2$$ is equivalent to the equation $$( 4 \cos \theta - 3 ) ( 2 \cos \theta + 1 ) = 0$$ 8
2. (ii) Solve the equation $$3 \sin \theta \tan \theta = 5 \cos \theta - 2$$ for \(- 180 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\) 8
3. Hence, deduce all the solutions of the equation $$3 \sin \left( \frac { 1 } { 2 } \theta \right) \tan \left( \frac { 1 } { 2 } \theta \right) = 5 \cos \left( \frac { 1 } { 2 } \theta \right) - 2$$ for \(- 180 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\), giving your answers to the nearest degree.
   [0pt] [2 marks]

4

1. 1. By using a suitable trigonometric identity, show that the equation $$\sin \theta \tan \theta = 4 \cos \theta$$ can be written as $$\tan ^ { 2 } \theta = 4$$ 4
2. (ii) Hence solve the equation $$\sin \theta \tan \theta = 4 \cos \theta$$ where \(0 ^ { \circ } < \theta < 360 ^ { \circ }\) Give your answers to the nearest degree.
   4
3. Deduce all solutions of the equation $$\sin 3 \alpha \tan 3 \alpha = 4 \cos 3 \alpha$$ where \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\) Give your answers to the nearest degree.

8

1. Given that $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ show that $$8 \cot ^ { 2 } \theta - 2 \cot \theta - 1 = 0$$ 8
2. Hence, solve $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ in the interval \(0 < \theta < 2 \pi\) Give your answers to two decimal places.
   8
3. Solve $$9 \sin ^ { 2 } \left( 2 x - \frac { \pi } { 4 } \right) + \sin \left( 4 x - \frac { \pi } { 2 } \right) = 8$$ in the interval \(0 < x < \frac { \pi } { 2 }\) Give your answers to one decimal place.