Multiple independent equations

8

1. It is given that \(\beta\) is an angle between \(90 ^ { \circ }\) and \(180 ^ { \circ }\) such that \(\sin \beta = a\).
   Express \(\tan ^ { 2 } \beta - 3 \sin \beta \cos \beta\) in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-11_2726_35_97_20}
2. Solve the equation \(\sin ^ { 2 } \theta + 2 \cos ^ { 2 } \theta = 4 \sin \theta + 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

4

1. Express the equation \(3 \sin \theta = \cos \theta\) in the form \(\tan \theta = k\) and solve the equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
2. Solve the equation \(3 \sin ^ { 2 } 2 x = \cos ^ { 2 } 2 x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

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

1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$3 \sin x + 7 \cos x = 0$$ Give each solution, in degrees, to one decimal place.
2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$10 \cos ^ { 2 } \theta + \cos \theta = 11 \sin ^ { 2 } \theta - 9$$ Give each solution, in radians, to 3 significant figures.

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.

5. (In this question, solutions based entirely on graphical or numerical methods are not acceptable.)

1. Solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$5 \sin 3 \theta - 7 \cos 3 \theta = 0$$ Give each solution, in radians, to 3 significant figures.
2. Solve, for \(0 < x < 360 ^ { \circ }\) $$9 \cos ^ { 2 } x + 5 \cos x = 3 \sin ^ { 2 } x$$ Give each solution, in degrees, to one decimal place.

12. [In this question solutions based entirely on graphical or numerical methods are not acceptable.]

1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$5 \sin \left( x + 65 ^ { \circ } \right) + 2 = 0$$ giving your answers in degrees to one decimal place.
2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$12 \sin ^ { 2 } \theta + \cos \theta = 6$$ giving your answers in radians to 3 significant figures.

7. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation $$3 \sin \left( 2 x - 15 ^ { \circ } \right) = \cos \left( 2 x - 15 ^ { \circ } \right)$$ giving your answers to one decimal place.
2. Solve, for \(0 < \theta < 2 \pi\), the equation $$4 \sin ^ { 2 } \theta + 8 \cos \theta = 3$$ giving your answers to 3 significant figures.

1. In this question you must show all stages of your working.

Solutions based entirely on calculator technology are not acceptable.

1. Solve, for \(- \frac { \pi } { 2 } < x < \pi\), the equation $$5 \sin ( 3 x + 0.1 ) + 2 = 0$$ giving your answers, in radians, to 2 decimal places.
2. Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation $$2 \tan \theta \sin \theta = 5 + \cos \theta$$ giving your answers, in degrees, to one decimal place.

10. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\) $$\tan ^ { 2 } \left( 2 x + \frac { \pi } { 4 } \right) = 3$$
2. Solve, for \(0 < \theta < 360 ^ { \circ }\) $$( 2 \sin \theta - \cos \theta ) ^ { 2 } = 1$$ giving your answers, as appropriate, to one decimal place.

1. (a) Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which

$$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3$$ (b) Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which $$\tan ^ { 2 } \theta = 4$$

5. Solve, for \(0 \leqslant x \leqslant 180 ^ { \circ }\), the equation

1. \(\quad \sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }\),
2. \(\cos 2 x = - 0.9\), giving your answers to 1 decimal place.

9. Solve, for \(0 \leqslant x < 360 ^ { \circ }\),

1. \(\quad \sin \left( x - 20 ^ { \circ } \right) = \frac { 1 } { \sqrt { 2 } }\)
2. \(\cos 3 x = - \frac { 1 } { 2 }\)

7. (i) Solve, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), $$( 1 + \tan \theta ) ( 5 \sin \theta - 2 ) = 0$$ (ii) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \sin x = 3 \tan x .$$

1. (a) Solve for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers in degrees to 1 decimal place,

$$3 \sin \left( x + 45 ^ { \circ } \right) = 2$$ (b) Find, for \(0 \leqslant x < 2 \pi\), all the solutions of $$2 \sin ^ { 2 } x + 2 = 7 \cos x$$ giving your answers in radians.
You must show clearly how you obtained your answers.

1. (i) Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\)

$$\sin \left( 2 \theta - 30 ^ { \circ } \right) + 1 = 0.4$$ giving your answers to 1 decimal place.
(ii) Find all the values of \(x\), in the interval \(0 \leqslant x < 360 ^ { \circ }\), for which $$9 \cos ^ { 2 } x - 11 \cos x + 3 \sin ^ { 2 } x = 0$$ giving your answers to 1 decimal place. You must show clearly how you obtained your answers.

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.)

7. (i) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$9 \sin \left( \theta + 60 ^ { \circ } \right) = 4$$ giving your answers to 1 decimal place.
You must show each step of your working.
(ii) Solve, for \(- \pi \leqslant x < \pi\), the equation $$2 \tan x - 3 \sin x = 0$$ giving your answers to 2 decimal places where appropriate. [Solutions based entirely on graphical or numerical methods are not acceptable.]

8. (i) Solve, for \(0 \leqslant \theta < \pi\), the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of \(\pi\).
(ii) Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$

1. find \(\cos x\) in terms of \(k\).
2. When \(k = 3\), find the values of \(x\) in the range \(0 \leqslant x < 360 ^ { \circ }\)

6. (i) Solve, for \(- \pi < \theta \leqslant \pi\), $$1 - 2 \cos \left( \theta - \frac { \pi } { 5 } \right) = 0$$ giving your answers in terms of \(\pi\).
(ii) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \cos ^ { 2 } x + 7 \sin x - 2 = 0$$ giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

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

1. Solve for \(0 \leqslant x < 360 ^ { \circ }\), $$4 \cos \left( x + 70 ^ { \circ } \right) = 3$$ giving your answers in degrees to one decimal place.
2. Find, for \(0 \leqslant \theta < 2 \pi\), all the solutions of $$6 \cos ^ { 2 } \theta - 5 = 6 \sin ^ { 2 } \theta + \sin \theta$$ giving your answers in radians to 3 significant figures.

1. (a) Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation \(\cos \left( x - 20 ^ { \circ } \right) = - 0.437\), giving your answers to the nearest degree.
   (b) Find the exact values of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which

$$3 \tan \theta = 2 \cos \theta$$

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

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

5. (i) Given that \(\sin \theta = 2 - \sqrt { 2 }\), find the value of \(\cos ^ { 2 } \theta\) in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
(ii) Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos 3 x = \frac { \sqrt { 3 } } { 2 }$$

8. (i) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\tan 2 x = 3$$ (ii) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y$$