Reduce to quadratic in trig

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 \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

11. In this question solutions based entirely on graphical or numerical methods are not acceptable.

1. Solve, for \(0 \leqslant x < 2 \pi\), $$3 \cos ^ { 2 } x + 1 = 4 \sin ^ { 2 } x$$ giving your answers in radians to 2 decimal places.
2. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$5 \sin \left( \theta + 10 ^ { \circ } \right) = \cos \left( \theta + 10 ^ { \circ } \right)$$ giving your answers in degrees to one decimal place.

7. (i) Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$\frac { \sin 2 \theta } { ( 4 \sin 2 \theta - 1 ) } = 1$$ giving your answers to 1 decimal place.
(ii) Solve, for \(0 \leqslant x < 2 \pi\), the equation $$5 \sin ^ { 2 } x - 2 \cos x - 5 = 0$$ giving your answers to 2 decimal places. (Solutions based entirely on graphical or numerical methods are not acceptable.)

5 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(2 \sin ^ { 2 } x = 1 + \cos x\).
2. \(\sin 2 x = - \cos 2 x\).

4. $$\mathrm { f } ( x ) = \frac { 4 } { 2 + \sin x ^ { \circ } }$$

1. State the maximum value of \(\mathrm { f } ( x )\) and the smallest positive value of \(x\) for which \(\mathrm { f } ( x )\) takes this value.
2. Solve the equation \(\mathrm { f } ( x ) = 3\) for \(0 \leq x \leq 360\), giving your answers to 1 decimal place.

5 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(\sin 2 x = 0.5\)
2. \(2 \sin ^ { 2 } x = 2 - \sqrt { 3 } \cos x\)

4 In this question you must show detailed reasoning. Solve the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. \(2 \tan x + 1 = 4\)
2. \(5 \sin x - 1 = 2 \cos ^ { 2 } x\)

4

1. Solve the equation \(\sec x = \frac { 3 } { 2 }\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
2. By using a suitable trigonometrical identity, solve the equation $$2 \tan ^ { 2 } x = 10 - 5 \sec x$$ giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

3

1. Solve the equation $$\operatorname { cosec } x = 3$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
   (2 marks)
2. By using a suitable trigonometric identity, solve the equation $$\cot ^ { 2 } x = 11 - \operatorname { cosec } x$$ giving all values of \(x\) in radians to two decimal places, in the interval \(0 \leqslant x \leqslant 2 \pi\).
   (6 marks)