Question

How do you solve $\cos \left( {2x} \right) = \dfrac{1}{2}$ and find all exact general solutions?

Answer 1

Hint: The given trigonometric equation can be solved by using the basic trigonometric properties of its ratios and angles. We will use the cosine formula of trigonometry ratios to convert the cosine function in its simple form cos. We will find the value of $x$ in the range of cosine function so we will use the formula of $\cos 2x$and simplify it or we can also solve the given trigonometric equation by comparison method. In the comparison method we compare the value of cosine angles and equate it with each other. We will use different trigonometric ratios and their properties to solve the equation. We will standardize the table of trigonometric ratios values at different angles.

Complete step by step answer:
Step: 1 the given equation of the equation is $\cos 2x = \dfrac{1}{2}$. We will have to find the value of $x$ and exact solutions of the equation.
Use the cosine formula to simplify the equation.
$
   \Rightarrow \cos 2x = \dfrac{1}{2} \\
   \Rightarrow 2{\cos ^2}x - 1 = \dfrac{1}{2} \\
 $
 Step: 2 we can also solve the equation by comparison method.
Assume $2x = \theta $ in the given equation.
$
   \Rightarrow \cos 2x = \dfrac{1}{2} \\
   \Rightarrow \cos \theta = \dfrac{1}{2} \\
 $
Now substitute the value of angle of $\theta $ at which the $\cos \theta = \dfrac{1}{2}$.
So the general solutions for the equation $\cos \theta = \dfrac{1}{2}$ are,
$\theta = \left\{ {\dfrac{\pi }{2} + 2n\pi , - \dfrac{\pi }{2} + 2n\pi } \right\}$
Substitute the value of $\theta = 2x$ in the general solution equation.
$
   \Rightarrow \cos 2x = \dfrac{1}{2} \\
  2x = \left\{ {\dfrac{\pi }{2} + 2n\pi , - \dfrac{\pi }{2} + 2n\pi } \right\} \\
 $
Solve the value for the $x$.
$x = \left( {\dfrac{\pi }{4} + n\pi , - \dfrac{\pi }{4} + n\pi } \right)$
Where $n$ belongs to the whole number.

Therefore the general solution of the equation is $x = \left( {\dfrac{\pi }{4} + n\pi , - \dfrac{\pi }{4} + n\pi } \right)$.

Note: Use the general solution formula to solve the trigonometry equation. Use the comparison method to find the solution of the equation. Assume $2x = \theta $ and solve for the $\theta $. Find the value of angles so that it satisfies the given equation.