Question

How do you solve $\cos \left( x \right)=\dfrac{1}{2}$ over the interval $0$ to $2\pi ?$

Answer 1

Hint: The given problem is related to the trigonometric equation. To solve a trigonometric equation so we have to transform it into one or many basic trigonometric equations. The solving a trigonometric equation, finally results in solving various basic trigonometric equations. There are four main basic trigonometric equations: $\sin x=a;\cos x=a;\tan x=a,\cot x=a$. For this problem we know the trigonometric values.

Complete step-by-step answer:
We have,
$\Rightarrow$ $\cos \left( x \right)=\dfrac{1}{2}$
We know that cosine is $\dfrac{1}{2}$. When we have $x=\dfrac{\pi }{3}$ however, the cosine is also positive on the fourth quadrant.
Now, we have to find another possible value of $x$ is.
$x=2\pi -\dfrac{\pi }{3}$
$=\dfrac{6\pi -\pi }{3}$
$=\dfrac{5\pi }{3}$

After solving $\cos \left( x \right)=\dfrac{1}{2}$ we get the value are $\dfrac{\pi }{3}$ and $\dfrac{5\pi }{3}$

Additional Information:
As a general description, there are the three steps that may be challenging or even impossible, depending on equation:
(1) Find the trigonometric values needed to solve the equation.
(2) Find all the angles that give us these values from step one.
(3) And then find the values of unknown that will result in angles that we get from step (2)
These are the important steps to find the trigonometric equation.
Here we take some examples of trigonometric expression which are as follows:
$f(x)={{\sin }^{2}}x+\cos x;$
$f(x)=\sin x+\sin 2x+\sin 3$
Use the trigonometric identities to transform the above given expression in a simplest for,
As an example take the above equation:
$f(x)=\sin 2x+\cos x$
Use trigonometric identities $(\sin 2a=2\sin a.\cos a)$ to transform $f(x)$
Then the expression becomes.
$f(x)=2.\sin x\cos x+\cos x=\cos x\left( 2\sin x+1 \right)$
This $f(x)$ expressed in the simplest form.

Note:
Look for a pattern that shows an algebraic property, such as the difference of a square or a factoring opportunity. Another point you have to remember is that substitute the trigonometric expression with a single variable as an example take $'x'$ or $'u'$ Not all the functions can be solved exactly using only the unit circle. When we must solve any equation involving an angle other than one of special sure the cases should be other modes should depend on the type or criteria of the given problem.