General solution — then find specific solutions

Find the general solution and then use it to identify specific solutions satisfying an additional condition, such as solutions in a given interval, the smallest solution greater than a value, or the solution closest to a given value.

6 questions · Moderate -0.1

1.05o Trigonometric equations: solve in given intervals

Sort by: Default | Easiest first | Hardest first

AQA FP1 2008 June Q5

7 marks Moderate -0.3

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 $\pi$.
Hence find the smallest positive value of $x$ which satisfies this equation.

AQA FP1 2009 June Q5

9 marks Standard +0.3

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

AQA FP1 2011 June Q5

7 marks Moderate -0.3

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 $\pi$.
Use your general solution to find the smallest solution of this equation which is greater than $5 \pi$.

AQA FP1 2015 June Q4

6 marks Moderate -0.3

Find the general solution, in degrees, of the equation $$2 \sin \left( 3 x + 45 ^ { \circ } \right) = 1$$
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]

AQA FP1 2014 June Q8

9 marks Standard +0.3

Find the general solution of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ giving your answer for $x$ in terms of $\pi$. [5 marks]
Use your general solution to find the sum of all the solutions of the equation $$\cos\left(\frac{5}{4}x - \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2}$$ that lie in the interval $0 \leqslant x \leqslant 20\pi$. Give your answer in the form $k\pi$, stating the exact value of $k$. [4 marks]

AQA FP1 2016 June Q4

7 marks Moderate -0.3

Given that $\sin \frac{\pi}{3} = \cos \frac{\pi}{k}$, state the value of the integer $k$. [1 mark]
Hence, or otherwise, find the general solution of the equation $$\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}$$ giving your answer, in its simplest form, in terms of $\pi$. [4 marks]
Hence, given that $\cos \left( 2x - \frac{5\pi}{6} \right) = \sin \frac{\pi}{3}$, show that there is only one finite value for $\tan x$ and state its exact value. [2 marks]