Show equation reduces to tan form

2

1. Express the equation \(\sin 2 x + 3 \cos 2 x = 3 ( \sin 2 x - \cos 2 x )\) in the form \(\tan 2 x = k\), where \(k\) is a constant.
2. Hence solve the equation for \(- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).

1

1. Show that the equation $$\sin \left( x - 60 ^ { \circ } \right) - \cos \left( 30 ^ { \circ } - x \right) = 1$$ can be written in the form \(\cos x = k\), where \(k\) is a constant.
2. Hence solve the equation, for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

5 The angle \(x\), measured in degrees, satisfies the equation $$\cos \left( x - 30 ^ { \circ } \right) = 3 \sin \left( x - 60 ^ { \circ } \right)$$

1. By expanding each side, show that the equation may be simplified to $$( 2 \sqrt { } 3 ) \cos x = \sin x$$
2. Find the two possible values of \(x\) lying between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
3. Find the exact value of \(\cos 2 x\), giving your answer as a fraction.

4

1. Show that the equation $$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$ can be written in the form $$( 3 \sqrt { } 3 ) \sin x = \cos x$$
2. Hence solve the equation $$\sin \left( x + 30 ^ { \circ } \right) = 2 \cos \left( x + 60 ^ { \circ } \right)$$ for \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

4

1. Show that the equation \(\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x\) can be written in the form \(\tan x = k\), where \(k\) is a constant.
2. Hence solve the equation \(\sin \left( 60 ^ { \circ } - x \right) = 2 \sin x\), for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

3

1. Given that \(\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)\), show that \(\tan x = \frac { 2 - \sqrt { 3 } } { 1 - 2 \sqrt { 3 } }\).
2. Hence solve the equation $$\cos \left( x - 30 ^ { \circ } \right) = 2 \sin \left( x + 30 ^ { \circ } \right)$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

4

1. Show that the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) can be expressed in the form $$\cos ^ { 2 } \theta + 2 \sin \theta \cos \theta - 3 \sin ^ { 2 } \theta = 0$$
2. Hence solve the equation \(\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

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

1. Solve, for \(0 < \theta < 360 ^ { \circ }\), the equation $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 7 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers to one decimal place.
2. (a) Show that the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ can be written in the form $$\sin x \left( a \cos ^ { 2 } x + b \cos x + c \right) = 0$$ where \(a , b\) and \(c\) are constants to be found.
   (b) Hence solve for \(- \frac { \pi } { 2 } \leqslant x \leqslant \frac { \pi } { 2 }\) the equation $$3 \sin ^ { 3 } x = 5 \sin x - 7 \sin x \cos x$$ \includegraphics[max width=\textwidth, alt={}, center]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-27_2644_1840_118_111}

2. (a) Show that the equation $$8 \cos \theta = 3 \operatorname { cosec } \theta$$ can be written in the form $$\sin 2 \theta = k$$ where \(k\) is a constant to be found.
(b) Hence find the smallest positive solution of the equation $$8 \cos \theta = 3 \operatorname { cosec } \theta$$ giving your answer, in degrees, to one decimal place.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$\frac { 3 \sin \theta \cos \theta } { \cos \theta + \sin \theta } = ( 2 + \sec 2 \theta ) ( \cos \theta - \sin \theta )$$ can be written in the form $$3 \sin 2 \theta - 4 \cos 2 \theta = 2$$
2. Hence solve for \(\pi < x < \frac { 3 \pi } { 2 }\) $$\frac { 3 \sin x \cos x } { \cos x + \sin x } = ( 2 + \sec 2 x ) ( \cos x - \sin x )$$ giving the answer to 3 significant figures.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Given that $$\sqrt { 2 } \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 60 ^ { \circ } \right)$$ show that $$\tan x = - 2 - \sqrt { 3 }$$
2. Hence or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$\sqrt { 2 } \sin ( 2 \theta ) = \cos \left( 2 \theta - 105 ^ { \circ } \right)$$

4. In this question you should show detailed reasoning. \section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Show that the equation $$2 \sin \left( \theta - 30 ^ { \circ } \right) = 5 \cos \theta$$ can be written in the form $$\tan \theta = 2 \sqrt { 3 }$$
2. Hence, or otherwise, solve for \(0 \leqslant x \leqslant 360 ^ { \circ }\) $$2 \sin \left( x - 10 ^ { \circ } \right) = 5 \cos \left( x + 20 ^ { \circ } \right)$$ giving your answers to one decimal place.

7. (a) Given that $$2 \cos ( x + 30 ) ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ without using a calculator, show that $$\tan x ^ { \circ } = 3 \sqrt { 3 } - 4$$ (b) Hence or otherwise solve, for \(0 \leqslant \theta < 180\), $$2 \cos ( 2 \theta + 40 ) ^ { \circ } = \sin ( 2 \theta - 20 ) ^ { \circ }$$ Give your answers to one decimal place.

3. Given that $$2 \cos ( x + 50 ) ^ { \circ } = \sin ( x + 40 ) ^ { \circ }$$

1. Show, without using a calculator, that $$\tan x ^ { \circ } = \frac { 1 } { 3 } \tan 40 ^ { \circ }$$
2. Hence solve, for \(0 \leqslant \theta < 360\), $$2 \cos ( 2 \theta + 50 ) ^ { \circ } = \sin ( 2 \theta + 40 ) ^ { \circ }$$ giving your answers to 1 decimal place.

1. (i) Show that the equation

$$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ can be written as $$\sqrt { 3 } \sin x \cos x + \cos ^ { 2 } x = 0$$ (ii) Hence, or otherwise, find in terms of \(\pi\) the solutions of the equation $$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ for \(x\) in the interval \(0 \leq x \leq \pi\).

4 The angle \(\theta\) satisfies the equation \(\sin \left( \theta + 45 ^ { \circ } \right) = \cos \theta\).

1. Using the exact values of \(\sin 45 ^ { \circ }\) and \(\cos 45 ^ { \circ }\), show that \(\tan \theta = \sqrt { 2 } - 1\).
2. Find the values of \(\theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

5 Given that \(\sin ( \theta + \alpha ) = 2 \sin \theta\), show that \(\tan \theta = \frac { \sin \alpha } { 2 - \cos \alpha }\). Hence solve the equation \(\sin \left( \theta + 40 ^ { \circ } \right) = 2 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

2．（a）Show that the equation $$\tan x = \frac { \sqrt { 3 } } { 1 + 4 \cos x }$$ can be written in the form $$\sin 2 x = \sin \left( 60 ^ { \circ } - x \right)$$ （b）Solve，for \(0 < x < 180 ^ { \circ }\) $$\tan x = \frac { \sqrt { 3 } } { 1 + 4 \cos x }$$

6 It is given that the angle \(\theta\) satisfies the equation \(\sin \left( 2 \theta + \frac { 1 } { 4 } \pi \right) = 3 \cos \left( 2 \theta + \frac { 1 } { 4 } \pi \right)\).

1. Show that \(\tan 2 \theta = \frac { 1 } { 2 }\).
2. Hence find, in surd form, the exact value of \(\tan \theta\), given that \(\theta\) is an obtuse angle.

8

1. Show that \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = \sin 2 \theta\).
2. In this question you must show detailed reasoning. Solve \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = 3 \cos 2 \theta\) for \(0 \leq \theta \leq \pi\).

8

1. Show that \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = \sin 2 \theta\).
2. In this question you must show detailed reasoning. Solve \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = 3 \cos 2 \theta\) for \(0 \leq \theta \leq \pi\).