Question
Question: How do you solve \( \cos 2x = \cos x \) ?...
How do you solve cos2x=cosx ?
Solution
In order to determine the solution of the above trigonometric equation, Rewrite the equation using trigonometric identity cos2x=2cos2x−1 . You will obtain a quadratic equation, compare it with the standard equation to get the value of variables a,b,c . Since a+b+c=0 , obtain the solution by putting cosx=0 and cosx=ac=−21 .
Complete step by step answer:
We are given a trigonometric equation cos2x=cosx and we have to find its solution
Since in the question we have not given any constraint for the value of x , so we will be finding the solution from 0 to 2π
We are going to rewrite our using the identity of trigonometry cos2x=2cos2x−1
2cos2x−1=cosx 2cos2x−cosx−1=0
Let X=cosx
As we can see we obtained a quadratic equation, lets compare the above equation with the standard quadratic equation aX2+bX+c=0 , we get
a=2 b=−1 c=−1
Since a+b+c=2−1−1=0 , we can say that there are two real roots as
a. cosx=0 , ⇒x=cos−1(0) ⇒x=0or2π
b. cosx=ac=−21 , ⇒x=cos−1(−21)⇒x=32πor34π
Therefore, the solution of given trigonometric equation is x=0,32π,34π,2π between 0 to 2π .
Additional Information:
1. Trigonometry is one of the significant branches throughout the entire existence of mathematics and this idea is given by a Greek mathematician Hipparchus.
2. Even Function – A function f(x) is said to be an even function ,if f(−x)=f(x) for all x in its domain.
Odd Function – A function f(x) is said to be an even function ,if f(−x)=−f(x) for all x in its domain.
We know that sin(−θ)=−sinθ.cos(−θ)=cosθandtan(−θ)=−tanθ
Therefore, sinθ and tanθ their reciprocals, cosecθ and cotθ are odd functions whereas cosθ and its reciprocal secθ are even functions.
3. Periodic Function= A function f(x) is said to be a periodic function if there exists a real number T > 0 such that f(x+T)=f(x) for all x.
Note: 1.One must be careful while taking values from the trigonometric table and cross-check at least once to avoid any error in the answer.
2. Period of cosine function is 2π .
3. The domain of cosine function is in the interval [0,π] and the range is in the interval [−1,1] .