Question

How do you solve $\cos \left( {2t} \right) = \dfrac{1}{2}$?

Answer 1

Hint: First assume \[2t\] as \[x\]. After that find the general solution of $\cos x = \dfrac{1}{2}$. Then substitute back the value of $x$ and divide both sides by 2 to get the value of the answer. A general solution is a value of $x$ which is true for all numbers in a certain range. We will simplify the equation using algebraic and trigonometric identities, such that the left-hand side and right-hand side are the sines of some angle. Then, using the trigonometric equations, we will find the general solution of the equation.

Complete step-by-step answer:
First, substitute $x$ in place of $2t$.
$ \Rightarrow \cos x = \dfrac{1}{2}$
We know that $\cos \dfrac{\pi }{3} = \dfrac{1}{2}$
$ \Rightarrow \cos x = \cos \dfrac{\pi }{3}$
Now, we will use the general solution of cosine.
So, we have $\cos \theta = \cos \alpha $, then the general solution is,
$\theta = 2n\pi \pm \alpha $
Now, using the above formula, we will get,
$ \Rightarrow x = 2n\pi \pm \dfrac{\pi }{3}$
Now, substitute back the value of $x$,
$ \Rightarrow 2t = 2n\pi \pm \dfrac{\pi }{3}$
Divide both sides by 2,
$ \Rightarrow t = n\pi \pm \dfrac{\pi }{6}$

Hence, the solution is $n\pi \pm \dfrac{\pi }{6}$.

Note:
Whenever we are stuck with these types of problems of finding general solutions, we always try to find the basic angle from the value obtained by algebraic operations. And then use the quadrant rule in trigonometry to get the general solution.
The solutions of a trigonometric equation for which $0 \le x \le 2\pi $ are called principal solutions. The mathematical expression involving integer n which gives all solutions of a trigonometric equation is called the general solution. In this type of problem, we must remember the values of trigonometric functions for particular angles. Also, we must remember some trigonometric identities, results, and formulas.
In the first quadrant, all are positive.
In the second quadrant, sine and cosecant (inverse of sine) are positive. Rest all are negative.
In the third quadrant, tangent and cotangent (inverse of tangent) are positive. Rest all are negative.
In the fourth quadrant, cosine and secant (inverse of the cosine) are positive. Rest all are negative.