Question

The general solution of the equation, $2\cos 2x=3.2{{\cos }^{2}}x-4$is: -
(a) $x=2n\pi ,n\in I$
(b) $x=n\pi ,n\in I$
(c) $x=\dfrac{n\pi }{4},n\in I$
(d) $x=\dfrac{n\pi }{2},n\in I$

Answer 1

Convert into its half angle by using the identity, $\cos 2x=2{{\cos }^{2}}x-1$. Form a quadratic equation in $\cos x$ and solve this equation to get the values of $\cos x$. If any value does not lie in the range of $\cos x$, i.e. [-1, 1] then eliminate that value. For the trigonometric equation as: - $x=n\pi \pm a,n\in I$.

Complete step-by-step solution
We have been provided with the equation: -
$\Rightarrow$ $2\cos 2x=3.2{{\cos }^{2}}x-4$.
First of all, note that in R.H.S the coefficient of ${{\cos }^{2}}x$ is 3.2. It is not a decimal number but it denotes 3 multiplied by 2.
So, the equation becomes,
$\Rightarrow 2\cos 2x=6{{\cos }^{2}}x-4$
Using the half angle formula to write $\cos 2x$ in terms of ${{\cos }^{2}}x$, we have,
$\cos 2x=2{{\cos }^{2}}x-1$, hence the required equation becomes,

& \Rightarrow 2\left( 2{{\cos }^{2}}x-1 \right)=6{{\cos }^{2}}x-4 \\\ & \Rightarrow 4{{\cos }^{2}}x-2=6{{\cos }^{2}}x-4 \\\ & \Rightarrow 2{{\cos }^{2}}x=2 \\\ & \Rightarrow {{\cos }^{2}}x=1 \\\ \end{aligned}$$ Now, we know that, $${{\cos }^{2}}0=1$$, therefore, $$\Rightarrow {{\cos }^{2}}x={{\cos }^{2}}0$$ The general solution for trigonometric equations of the form: - $${{\cos }^{2}}x={{\cos }^{2}}a$$ is given by $$x=n\pi \pm a,n\in I$$. $$\begin{aligned} & \Rightarrow {{\cos }^{2}}x={{\cos }^{2}}0 \\\ & \Rightarrow x=n\pi \pm 0 \\\ & \Rightarrow x=n\pi ,n\in I \\\ \end{aligned}$$ **Hence, option (B) is the correct answer.** **Note:** One may note that in the starting we have written 3.2 as 3 multiplied by 2. This is because in higher mathematics dot (.) is the symbol for multiplication. In the above question, we have converted $$\cos 2x$$ and written it in term of $${{\cos }^{2}}x$$, but one can also convert $${{\cos }^{2}}x$$ in the R.H.S into $$\cos 2x$$ by using the formula: - $${{\cos }^{2}}x=\dfrac{1+\cos 2x}{2}$$. So, at last we will have a equation of the form, $$\cos 2x=\cos a$$ whose general solution is given as: - $$2x=2n\pi +a,n\in I$$.