General solution

3 Find the general solution, in degrees, for the equation $$\sin \left( 4 x + 10 ^ { \circ } \right) = \sin 50 ^ { \circ }$$

3 Find the general solution of the equation $$\tan \left( \frac { \pi } { 2 } - 3 x \right) = \sqrt { 3 }$$

4 Find the general solution of the equation $$\sin \left( 4 x - \frac { 2 \pi } { 3 } \right) = - \frac { 1 } { 2 }$$ giving your answer in terms of \(\pi\).
(6 marks)

6 Find the general solution of each of the following equations:

1. \(\quad \tan \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right) = \frac { 1 } { \sqrt { 3 } }\);
2. \(\quad \tan ^ { 2 } \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right) = \frac { 1 } { 3 }\).

6 Find the general solution of the equation $$\sin \left( 2 x - \frac { \pi } { 2 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answer in terms of \(\pi\).

5

1. Find, in radians, the general solution of the equation $$\cos \left( \frac { x } { 2 } + \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$ giving your answer in terms of \(\pi\).
2. Hence find the smallest positive value of \(x\) which satisfies this equation.

5

1. Find the general solution of the equation $$\cos ( 3 x - \pi ) = \frac { 1 } { 2 }$$ giving your answer in terms of \(\pi\).
2. From your general solution, find all the solutions of the equation which lie between \(10 \pi\) and \(11 \pi\).

5

1. Find the general solution of the equation $$\cos \left( 3 x - \frac { \pi } { 6 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answer in terms of \(\pi\).
2. Use your general solution to find the smallest solution of this equation which is greater than \(5 \pi\).

4 Find the general solution, in degrees, of the equation $$\sin \left( 70 ^ { \circ } - \frac { 2 } { 3 } x \right) = \cos 20 ^ { \circ }$$

3

1. Find the general solution, in degrees, of the equation $$\cos \left( 5 x + 40 ^ { \circ } \right) = \cos 65 ^ { \circ }$$
2. Given that $$\sin \frac { \pi } { 12 } = \frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$$ express \(\sin \frac { \pi } { 12 }\) in the form \(\left( \cos \frac { \pi } { 4 } \right) ( \cos ( a \pi ) + \cos ( b \pi ) )\), where \(a\) and \(b\) are rational.
   (3 marks)

4

1. Find the general solution, in degrees, of the equation $$2 \sin \left( 3 x + 45 ^ { \circ } \right) = 1$$
2. Use your general solution to find the solution of \(2 \sin \left( 3 x + 45 ^ { \circ } \right) = 1\) that is closest to \(200 ^ { \circ }\).
   [0pt] [1 mark]

3 Find the general solution of the equation $$\tan 4 \left( x - \frac { \pi } { 8 } \right) = 1$$ giving your answer in terms of \(\pi\).

3 Find the general solution of the equation $$\sin \left( 4 x + \frac { \pi } { 4 } \right) = 1$$

5 Find the general solutions of the following equations, giving your answers in terms of \(\pi\) :

1. \(\quad \tan 3 x = \sqrt { 3 }\);
2. \(\quad \tan \left( 3 x - \frac { \pi } { 3 } \right) = - \sqrt { 3 }\).

4 Find, in radians, the general solution of the equation $$\cos 3 x = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in terms of \(\pi\).