Show then solve: sin²/cos² substitution

6 The function \(\mathrm { f } : x \mapsto 5 \sin ^ { 2 } x + 3 \cos ^ { 2 } x\) is defined for the domain \(0 \leqslant x \leqslant \pi\).

1. Express \(\mathrm { f } ( x )\) in the form \(a + b \sin ^ { 2 } x\), stating the values of \(a\) and \(b\).
2. Hence find the values of \(x\) for which \(\mathrm { f } ( x ) = 7 \sin x\).
3. State the range of f .

7

1. \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-12_499_568_267_826} The diagram shows part of the graph of \(y = a + b \sin x\). Find the values of the constants \(a\) and \(b\).
2. 1. Show that the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ may be expressed as \(3 \cos ^ { 2 } \theta - 2 \cos \theta - 1 = 0\).
   2. Hence solve the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ for \(- 180 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8. In this question the angle \(\theta\) is measured in degrees throughout.

1. Show that the equation $$\frac { 5 + \sin \theta } { 3 \cos \theta } = 2 \cos \theta , \quad \theta \neq ( 2 n + 1 ) 90 ^ { \circ } , \quad n \in \mathbb { Z }$$ may be rewritten as $$6 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
2. Hence solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation $$\frac { 5 + \sin \theta } { 3 \cos \theta } = 2 \cos \theta$$ Give your answers to one decimal place, where appropriate.

12. (a) Show that $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ may be expressed in the form $$a \cos ^ { 2 } x + b \cos x + c = 0$$ where \(a , b\) and \(c\) are constants to be found.
(b) Hence solve, for \(0 \leqslant x < 2 \pi\), the equation $$\frac { 2 + \cos x } { 3 + \sin ^ { 2 } x } = \frac { 4 } { 7 }$$ giving your answers in radians to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-37_81_65_2640_1886}

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 \(0 \leqslant x < 360 ^ { \circ }\), the equation $$5 \cos ^ { 2 } x = 3 ( 1 + \sin x )$$ giving your answers to 1 decimal place where appropriate.

8. (a) Show that the equation $$4 \sin ^ { 2 } x + 9 \cos x - 6 = 0$$ can be written as $$4 \cos ^ { 2 } x - 9 \cos x + 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 720 ^ { \circ }\), $$4 \sin ^ { 2 } x + 9 \cos x - 6 = 0$$ giving your answers to 1 decimal place.

2. (a) Show that the equation $$5 \sin x = 1 + 2 \cos ^ { 2 } x$$ can be written in the form $$2 \sin ^ { 2 } x + 5 \sin x - 3 = 0$$ (b) Solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$2 \sin ^ { 2 } x + 5 \sin x - 3 = 0$$

1. (a) Show that the equation

$$3 \sin ^ { 2 } x + 7 \sin x = \cos ^ { 2 } x - 4$$ can be written in the form $$4 \sin ^ { 2 } x + 7 \sin x + 3 = 0$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$3 \sin ^ { 2 } x + 7 \sin x = \cos ^ { 2 } x - 4$$ giving your answers to 1 decimal place where appropriate.

8. (a) Show that the equation $$\cos ^ { 2 } x = 8 \sin ^ { 2 } x - 6 \sin x$$ can be written in the form $$( 3 \sin x - 1 ) ^ { 2 } = 2$$ (b) Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), $$\cos ^ { 2 } x = 8 \sin ^ { 2 } x - 6 \sin x$$ giving your answers to 2 decimal places.

5

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

8

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 \(180 ^ { \circ }\). Section B (36 marks)

7 Show that the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) may be written in the form $$4 \sin ^ { 2 } \theta - \sin \theta = 0$$ Hence solve the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

5

1. Express \(2 \sin ^ { 2 } \theta + 3 \cos \theta\) as a quadratic function of \(\cos \theta\).
2. Hence solve the equation \(2 \sin ^ { 2 } \theta + 3 \cos \theta = 3\), giving all values of \(\theta\) correct to the nearest degree in the range \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).

3. (i) Show that the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$ can be written as a quadratic equation in \(\sin \chi ^ { \circ }\).
(ii) Hence solve, for \(0 \leq x < 360\), the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$

1 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 \(2 \pi\).

8 Show that the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) may be written in the form $$4 \sin ^ { 2 } \theta - \sin \theta = 0$$ Hence solve the equation \(4 \cos ^ { 2 } \theta = 4 - \sin \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

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

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 \(360 ^ { \circ }\).

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 \(2 \pi\).

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 \(2 \sqrt { 2 }\).