Question
Question: How do you solve \({\cos ^2}x - \cos x = 0\) ?...
How do you solve cos2x−cosx=0 ?
Solution
In the question, a polynomial equation of one variable is given, it is a function of cosx . To make it easier to solve the given polynomial equation, we replace cosx in the given equation with any other variable. We know that the degree of a polynomial equation is the highest power of the variable used in the polynomial and also the number of roots of a polynomial equation is equal to its degree, so several methods like factorizing the equation or by a special formula called quadratic formula or by completing the square method or quadratic formula can be used to solve the given equation; we use other methods if we are not able to factorize the equation.
Complete step by step answer:
We have to solve the equation cos2x−cosx=0
Taking cosx=t, we get –
⇒t2−t=0
Now we see that this equation can be solved by simply taking t as common –
⇒t(t−1)=0 ⇒t=0,t−1=0 ⇒t=0,t=1
Putting the original value of t as cosx , we get –
⇒cosx=0,cosx=1
We know that cos0=0,cos90∘=1, so we get –
⇒cosx=cos0,cosx=cos90∘ ⇒x=0∘,x=90∘
Hence, the solution of the equation cos2x−cosx=0 is x=0∘ or x=90∘.
Note: In this question, we have just written the general solution, there can be infinitely many solutions to the questions like this. The general solution lies in the interval [0,2π) that is the value of x can be greater than or equal to zero but smaller than 2π so we take only these two values as the answer.