Convert to quadratic in sin/cos

10

1. Show that the equation \(2 \cos ^ { 2 } \theta + 7 \sin \theta = 5\) may be written in the form $$2 \sin ^ { 2 } \theta - 7 \sin \theta + 3 = 0$$
2. By factorising this quadratic equation, solve the equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\). [4]

3 Show that the equation \(4 \cos ^ { 2 } \theta = 1 + \sin \theta\) can be expressed as $$4 \sin ^ { 2 } \theta + \sin \theta - 3 = 0$$ Hence solve the equation for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4 Showing your method clearly, solve the equation $$5 \sin ^ { 2 } \theta = 5 + \cos \theta \quad \text { for } 0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ } .$$

1

1. Show that the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ can be expressed in the form $$2 \cos ^ { 2 } x + 5 \cos x - 3 = 0$$
2. Hence solve the equation $$2 \sin ^ { 2 } x = 5 \cos x - 1$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

4 Solve the equation $$4 \cos ^ { 2 } x + 7 \sin x - 7 = 0$$ giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

8 Showing your method clearly, solve the equation $$5 \sin ^ { 2 } \theta = 5 + \cos \theta \quad \text { for } 0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ } .$$

8 Show that the equation \(4 \cos ^ { 2 } \theta = 1 + \sin \theta\) can be expressed as $$4 \sin ^ { 2 } \theta + \sin \theta - 3 = 0 .$$ Hence solve the equation for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

7 Show that the equation \(\sin ^ { 2 } x = 3 \cos x - 2\) can be expressed as a quadratic equation in \(\cos x\) and hence solve the equation for values of \(x\) between 0 and \(2 \pi\).

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$4 \tan x = 5 \cos x$$ can be written as $$5 \sin ^ { 2 } x + 4 \sin x - 5 = 0$$
2. Hence solve, for \(0 < x \leqslant 360 ^ { \circ }\) $$4 \tan x = 5 \cos x$$ giving your answers to one decimal place.
3. Hence find the number of solutions of the equation $$4 \tan 3 x = 5 \cos 3 x$$ in the interval \(0 < x \leqslant 1800 ^ { \circ }\), explaining briefly the reason for your answer.

1. Solve, for \(360 ^ { \circ } \leqslant x < 540 ^ { \circ }\),

$$12 \sin ^ { 2 } x + 7 \cos x - 13 = 0$$ Give your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(5)

4 In this question you must show detailed reasoning.
Solve the equation \(6 \cos ^ { 2 } x + \sin x = 5\), giving all the roots in the interval \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

14 In this question you must show detailed reasoning.
Solve the equation \(5 - \cos \theta - 6 \sin ^ { 2 } \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\). Turn over for question 15

8

1. Given that \(2 \sin \theta = 7 \cos \theta\), find the value of \(\tan \theta\).
2. 1. Use an appropriate identity to show that the equation $$6 \sin ^ { 2 } x = 4 + \cos x$$ can be written as $$6 \cos ^ { 2 } x + \cos x - 2 = 0$$
   2. Hence solve the equation \(6 \sin ^ { 2 } x = 4 + \cos x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\), giving your answers to the nearest degree.

6

1. Solve the equation \(\sin ( x + 0.7 ) = 0.6\) in the interval \(- \pi < x < \pi\), giving your answers in radians to two significant figures.
2. It is given that \(5 \cos ^ { 2 } \theta - \cos \theta = \sin ^ { 2 } \theta\).
   1. By forming and solving a suitable quadratic equation, find the possible values of \(\cos \theta\).
   2. Hence show that a possible value of \(\tan \theta\) is \(2 \sqrt { 2 }\).

3. Find the values of \(\theta\), to 1 decimal place, in the interval \(- 180 \leq \theta < 180\) for which $$2 \sin ^ { 2 } \theta ^ { \circ } - 2 \sin \theta ^ { \circ } = \cos ^ { 2 } \theta ^ { \circ }$$ [P1 January 2002 Question 3]

6. Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which $$2 \cos ^ { 2 } \theta - \cos \theta - 1 = \sin ^ { 2 } \theta$$ Give your answers to 1 decimal place where appropriate.
(8)

4. Solve, for \(0 \leq x < 360\), the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$

1. Find all values of \(x\) in the interval \(0 \leq x < 360 ^ { \circ }\) for which

$$2 \sin ^ { 2 } x - 2 \cos x - \cos ^ { 2 } x = 1$$

4. Solve the equation $$\sin ^ { 2 } \theta = 4 \cos \theta ,$$ for values of \(\theta\) in the interval \(0 \leq \theta \leq 360 ^ { \circ }\).

2. Find all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\) satisfying $$7 \sin ^ { 2 } \theta + 1 = 3 \cos ^ { 2 } \theta - \sin \theta$$

2 In this question you must show detailed reasoning. Solve the equation \(2 \cos ^ { 2 } x = 2 - \sin x\) for \(0 ^ { \circ } \leq x \leq 180 ^ { \circ }\).

4. (a) Show that the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ can be written as $$5 \sin ^ { 2 } x + 3 \sin x - 2 = 0 .$$ (b) Hence solve, for \(0 \leq x < 360 ^ { \circ }\), the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x ) ,$$ giving your answers to 1 decimal place where appropriate.

10 Showing your method, solve the equation \(2 \sin ^ { 2 } \theta = \cos \theta + 2\) for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).