Express and solve equation

6

1. Express \(2 \sin \theta - \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$2 \sin \theta - \cos \theta = - 0.4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

8

1. Express \(5 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$5 \cos \theta - 3 \sin \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Write down the least value of \(15 \cos \theta - 9 \sin \theta\) as \(\theta\) varies.

3

1. Express \(8 \sin \theta + 15 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$8 \sin \theta + 15 \cos \theta = 6$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

6

1. Express \(( \sqrt { } 5 ) \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
2. Hence
   (a) solve the equation \(( \sqrt { } 5 ) \cos \theta - 2 \sin \theta = 0.9\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
   (b) state the greatest and least values of $$10 + ( \sqrt { } 5 ) \cos \theta - 2 \sin \theta$$ as \(\theta\) varies.

8

1. Express \(0.5 \cos \theta - 1.2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(0.5 \cos \theta - 1.2 \sin \theta = 0.8\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
3. Determine the greatest and least possible values of \(( 3 - \cos \theta + 2.4 \sin \theta ) ^ { 2 }\) as \(\theta\) varies.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3

1. Express \(8 \sin \theta + 15 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$8 \sin \theta + 15 \cos \theta = 6$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

5

1. Express \(\sqrt { 2 } \cos x - \sqrt { 5 } \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 3 decimal places.
2. Hence solve the equation \(\sqrt { 2 } \cos 2 \theta - \sqrt { 5 } \sin 2 \theta = 1\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

6

1. Express \(3 \cos x + 2 \cos \left( x - 60 ^ { \circ } \right)\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$3 \cos 2 \theta + 2 \cos \left( 2 \theta - 60 ^ { \circ } \right) = 2.5$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5

1. Express \(\sqrt { 7 } \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(\sqrt { 7 } \sin 2 \theta + 2 \cos 2 \theta = 1\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

6

1. Express \(5 \sin \theta + 12 \cos \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
2. Hence solve the equation \(5 \sin 2 x + 12 \cos 2 x = 6\) for \(0 \leqslant x \leqslant \pi\).

8

1. Express \(3 \sin x + 2 \sqrt { 2 } \cos \left( x + \frac { 1 } { 4 } \pi \right)\) in the form \(\mathrm { R } \sin ( \mathrm { x } + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). State the exact value of \(R\) and give \(\alpha\) correct to 3 decimal places.
2. Hence solve the equation $$6 \sin \frac { 1 } { 2 } \theta + 4 \sqrt { 2 } \cos \left( \frac { 1 } { 2 } \theta + \frac { 1 } { 4 } \pi \right) = 3$$ for \(- 4 \pi < \theta < 4 \pi\).

6

1. Express \(\sqrt { 6 } \cos \theta + 3 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(\sqrt { 6 } \cos \frac { 1 } { 3 } x + 3 \sin \frac { 1 } { 3 } x = 2.5\), for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

4

1. Express \(4 \cos x - \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(4 \cos 2 x - \sin 2 x = 3\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

9. (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 \(0 \leqslant x < 360 ^ { \circ }\)
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10. (a) Express \(3 \sin 2 x + 5 \cos 2 x\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) to 3 significant figures.
(b) Solve, for \(0 < x < \pi\), $$3 \sin 2 x + 5 \cos 2 x = 4$$ (Solutions based entirely on graphical or numerical methods are not acceptable.) $$g ( x ) = 4 ( 3 \sin 2 x + 5 \cos 2 x ) ^ { 2 } + 3$$ (c) Using your answer to part (a) and showing your working,

1. find the greatest value of \(\mathrm { g } ( x )\),
2. find the least value of \(\mathrm { g } ( x )\).

1. (a) Write \(\cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
   (b) Hence solve, for \(0 \leqslant \theta < \pi\), the equation

$$\cos 2 \theta + 4 \sin 2 \theta = 1.2$$ giving your answers to 2 decimal places.

1. (a) Express \(5 \cos 2 \theta - 12 \sin 2 \theta\) in the form \(R \cos ( 2 \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\) Give the value of \(\alpha\) to 2 decimal places.
   (b) Hence solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation

$$5 \cos 2 \theta - 12 \sin 2 \theta = 10$$ giving your answers to 1 decimal place.

6. $$f ( x ) = 12 \cos x - 4 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x + \alpha )\), where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\),

1. find the value of \(R\) and the value of \(\alpha\).
2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
3. 1. Write down the minimum value of \(12 \cos x - 4 \sin x\).
   2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs. \includegraphics[max width=\textwidth, alt={}, center]{5cd53af1-bac9-4ed9-ac45-59ad2e372423-09_60_35_2669_1853}

5. \begin{figure}[h] \end{figure} Figure 1 shows an oscilloscope screen. The curve shown on the screen satisfies the equation $$y = \sqrt { 3 } \cos x + \sin x$$

1. Express the equation of the curve in the form \(y = R \sin ( x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
2. Find the values of \(x , 0 \leqslant x < 2 \pi\), for which \(y = 1\).

3. (a) Express \(5 \cos x - 3 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(b) Hence, or otherwise, solve the equation $$5 \cos x - 3 \sin x = 4$$ for \(0 \leqslant x < 2 \pi\), giving your answers to 2 decimal places.

1. (a) Express \(7 \cos x - 24 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 decimal places.
   (b) Hence write down the minimum value of \(7 \cos x - 24 \sin x\).
   (c) Solve, for \(0 \leqslant x < 2 \pi\), the equation

$$7 \cos x - 24 \sin x = 10$$ giving your answers to 2 decimal places.

1. (a) Express \(3 \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
   (b) Hence find the greatest value of \(( 3 \sin x + 2 \cos x ) ^ { 4 }\).
   (c) Solve, for \(0 < x < 2 \pi\), the equation

$$3 \sin x + 2 \cos x = 1$$ giving your answers to 3 decimal places.

2. $$f ( x ) = 5 \cos x + 12 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\),

1. find the value of \(R\) and the value of \(\alpha\) to 3 decimal places.
2. Hence solve the equation $$5 \cos x + 12 \sin x = 6$$ for \(0 \leqslant x < 2 \pi\).
3. 1. Write down the maximum value of \(5 \cos x + 12 \sin x\).
   2. Find the smallest positive value of \(x\) for which this maximum value occurs.

1. (a) Use the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\), to show that

$$\cos 2 A = 1 - 2 \sin ^ { 2 } A$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 3 \sin 2 x \\ & C _ { 2 } : \quad y = 4 \sin ^ { 2 } x - 2 \cos 2 x \end{aligned}$$ (b) Show that the \(x\)-coordinates of the points where \(C _ { 1 }\) and \(C _ { 2 }\) intersect satisfy the equation $$4 \cos 2 x + 3 \sin 2 x = 2$$ (c) Express \(4 \cos 2 x + 3 \sin 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places.
(d) Hence find, for \(0 \leqslant x < 180 ^ { \circ }\), all the solutions of $$4 \cos 2 x + 3 \sin 2 x = 2$$ giving your answers to 1 decimal place.

3. $$f ( x ) = 7 \cos x + \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),

1. find the exact value of \(R\) and the value of \(\alpha\) to one decimal place.
2. Hence solve the equation $$7 \cos x + \sin x = 5$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
3. State the values of \(k\) for which the equation $$7 \cos x + \sin x = k$$ has only one solution in the interval \(0 \leqslant x < 360 ^ { \circ }\)