General solution — find all solutions

Find the general solution of a trigonometric equation (sin, cos, or tan of a linear expression in x), expressing all solutions in terms of n (integer), either in degrees or radians.

13 questions · Moderate -0.1

1.05o Trigonometric equations: solve in given intervals
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AQA FP1 2006 January Q3
5 marks Moderate -0.8
3 Find the general solution, in degrees, for the equation $$\sin \left( 4 x + 10 ^ { \circ } \right) = \sin 50 ^ { \circ }$$
AQA FP1 2009 January Q3
5 marks Moderate -0.5
3 Find the general solution of the equation $$\tan \left( \frac { \pi } { 2 } - 3 x \right) = \sqrt { 3 }$$
AQA FP1 2011 January Q4
6 marks Standard +0.3
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)
AQA FP1 2012 January Q6
7 marks Standard +0.3
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. \(\quad \tan ^ { 2 } \left( \frac { x } { 2 } - \frac { \pi } { 4 } \right) = \frac { 1 } { 3 }\).
AQA FP1 2007 June Q6
6 marks Moderate -0.3
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 \(\pi\).
AQA FP1 2012 June Q4
6 marks Moderate -0.3
4 Find the general solution, in degrees, of the equation $$\sin \left( 70 ^ { \circ } - \frac { 2 } { 3 } x \right) = \cos 20 ^ { \circ }$$
AQA FP1 2013 June Q3
8 marks Standard +0.3
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)
AQA FP1 2008 January Q3
5 marks Moderate -0.8
3 Find the general solution of the equation $$\tan 4 \left( x - \frac { \pi } { 8 } \right) = 1$$ giving your answer in terms of \(\pi\).
AQA FP1 2010 January Q3
4 marks Easy -1.2
3 Find the general solution of the equation $$\sin \left( 4 x + \frac { \pi } { 4 } \right) = 1$$
AQA FP1 2005 June Q5
7 marks Moderate -0.3
5 Find the general solutions of the following equations, giving your answers in terms of \(\pi\) :
  1. \(\quad \tan 3 x = \sqrt { 3 }\);
  2. \(\quad \tan \left( 3 x - \frac { \pi } { 3 } \right) = - \sqrt { 3 }\).
AQA FP1 2006 June Q4
5 marks Moderate -0.5
4 Find, in radians, the general solution of the equation $$\cos 3 x = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in terms of \(\pi\).
WJEC Further Unit 4 2023 June Q12
6 marks Challenging +1.2
Find the general solution of the equation $$\cos 4\theta + \cos 2\theta = \cos\theta.$$ [6]
WJEC Further Unit 4 2024 June Q9
9 marks Challenging +1.8
Find the general solution of the equation $$\sin 6\theta + 2\cos^2\theta = 3\cos 2\theta - \sin 2\theta + 1.$$ [9]