Multiple independent equations

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

1. \(3 \tan 2 x = 1\)
2. \(3 \cos ^ { 2 } x + 2 \sin x - 3 = 0\)

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

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

2 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. \(\sin \frac { 1 } { 2 } x = 0.8\)
2. \(\sin x = 3 \cos x\)

12. a. Explain mathematically why there are no values of \(\theta\) that satisfy the equation $$( 3 \cos \theta - 4 ) ( 2 \cos \theta + 5 ) = 0$$ b. Giving your solutions to one decimal place, where appropriate, solve the equation $$3 \sin y + 2 \tan y = 0 \quad \text { for } 0 \leq y \leq \pi$$ (Solutions based entirely on graphical or numerical methods are not acceptable.)

3

1. Solve the equation \(\sin ^ { 2 } \theta = 0.25\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
2. In this question you must show detailed reasoning. Solve the equation \(\tan 3 \phi = \sqrt { 3 }\) for \(0 ^ { \circ } \leqslant \phi < 90 ^ { \circ }\).

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

9

1. Solve the equation \(\sin 2 x = \sin 48 ^ { \circ }\), giving the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
2. Solve the equation \(2 \sin \theta - 3 \cos \theta = 0\) in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\), giving your answers to the nearest \(0.1 ^ { \circ }\).

3. Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which

1. \(\cos ( \theta + 75 ) ^ { \circ } = 0\).
2. \(\sin 2 \theta ^ { \circ } = 0.7\), giving your answers to one decima1 place.

3. (a) Given that $$5 \cos \theta - 2 \sin \theta = 0 ,$$ show that \(\tan \theta = 2.5\) (b) 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.

7. (a) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < 2 \pi\) for which $$\tan \left( x + \frac { \pi } { 4 } \right) = 3 .$$ (b) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y .$$

1. (a) Find the exact value of \(x\) such that

$$3 \arctan ( x - 2 ) + \pi = 0$$ (b) Solve, for \(- \pi < \theta < \pi\), the equation $$\cos 2 \theta - \sin \theta - 1 = 0$$ giving your answers in terms of \(\pi\).

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)