Question

If \[x = \dfrac{{n\pi }}{2}\] satisfies the equation \[\sin \dfrac{x}{2} - \cos \dfrac{x}{2} = 1 - \sin x\] and the inequality \[\left| {\dfrac{x}{2} - \dfrac{\pi }{2}} \right| \le \dfrac{{3\pi }}{4}\], then
A) \[n = - 1,0,3,5\]
B) \[n = 1,2,4,5\]
C) \[n = 0,2,4\]
D) \[n = - 1,1,3,5\]

Answer 1

Hint:
Here we will first find the values of \[x\] by solving the given inequality. Then we will substitute the values of \[x\] in the given equation. As we substitute those values, we will find the required values of \[n\].

Complete step by step solution:
At first we have to solve the inequality \[\left| {\dfrac{x}{2} - \dfrac{\pi }{2}} \right| \le \dfrac{{3\pi }}{4}\] to find the value of \[x\].
Here, modulus represents the non-negative integer without regard to its sign.
\[ \Rightarrow \left| {\dfrac{x}{2} - \dfrac{\pi }{2}} \right| \le \dfrac{{3\pi }}{4}\]
Taking modulus on both the sides, we get
\[ \Rightarrow \dfrac{x}{2} - \dfrac{\pi }{2} \le \left| {\dfrac{{3\pi }}{4}} \right|\]
\[ \Rightarrow \dfrac{{x - \pi }}{2} \le \dfrac{{3\pi }}{4}\]
Multiplying both side by 2, we get
\[ \Rightarrow x - \pi \le \dfrac{{3\pi }}{2}\]
Adding \[\pi \] on both the sides, we get
\[ \Rightarrow x \le \dfrac{{3\pi }}{2} + \pi \]
\[ \Rightarrow x \le \dfrac{{5\pi }}{2}\]
So, the value of \[x\] should be less than or equal to \[\dfrac{{5\pi }}{2}\] . Since the value lies in the modulus value, the negative value of \[n\] can be neglected. So the values of \[x\] are \[x = 0,\dfrac{\pi }{2},\dfrac{{2\pi }}{2},\dfrac{{3\pi }}{2},\dfrac{{4\pi }}{2},\dfrac{{5\pi }}{2}\] that is \[x = 0,\dfrac{\pi }{2},\pi ,\dfrac{{3\pi }}{2},2\pi ,\dfrac{{5\pi }}{2}\] .These values of \[x\] are obtained by the equation \[x = \dfrac{{n\pi }}{2}\].
These values of \[x\] are substituted in the equation \[\sin \dfrac{x}{2} - \cos \dfrac{x}{2} = 1 - \sin x\] to obtain the values of \[n\].
Substituting \[x = 0\] in the given equation, we get
\[\sin 0 - \cos 0 = 1 - \sin 0\]
Substituting the values of \[\sin 0\] and \[\cos 0\], we get
\[\begin{array}{l} \Rightarrow 0 - 1 = 1 - 0\\ \Rightarrow - 1 = 1\end{array}\]
\[x = 0\] does not satisfy the equation.
 Substituting \[x = \dfrac{\pi }{2}\] in the given equation, we get
\[\sin \dfrac{\pi }{4} - \cos \dfrac{\pi }{4} = 1 - \sin \dfrac{\pi }{2}\]
Substituting the values of \[\sin \dfrac{\pi }{2}\] and \[\cos \dfrac{\pi }{2}\], we get
\[\begin{array}{l} \Rightarrow \dfrac{1}{{\sqrt 2 }} - \dfrac{1}{{\sqrt 2 }} = 1 - 1\\ \Rightarrow 0 = 0\end{array}\]
\[x = \dfrac{\pi }{2}\] satisfies the equation.
Substituting \[x = \pi \] in the given equation, we get
\[\sin \dfrac{\pi }{2} - \cos \dfrac{\pi }{2} = 1 - \sin \pi \]
Substituting the values of \[\sin \pi \] and \[\cos \pi \], we get
\[\begin{array}{l} \Rightarrow 1 - 0 = 1 - 0\\ \Rightarrow 1 = 1\end{array}\]
\[x = \pi \] satisfies the equation.
Substitute \[x = \dfrac{{3\pi }}{2}\] in the given equation, we get
\[\sin \dfrac{{3\pi }}{4} - \cos \dfrac{{3\pi }}{4} = 1 - \sin \dfrac{{3\pi }}{2}\]
Substituting the values of \[\sin \dfrac{{3\pi }}{2}\] and \[\cos \dfrac{{3\pi }}{2}\], we get
\[\begin{array}{l} \Rightarrow \dfrac{1}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }} = 1 + 1\\ \Rightarrow \sqrt 2 = 2\end{array}\]
\[x = \dfrac{{3\pi }}{2}\]does not satisfy the equation.
Substituting \[x = 2\pi \] in the given equation, we get
\[\sin \pi - \cos \pi = 1 - \sin 2\pi \]
Substituting the values of \[\sin 2\pi \] and \[\cos 2\pi \], we get
\[\begin{array}{l} \Rightarrow 0 + 1 = 1 - 0\\ \Rightarrow 1 = 1\end{array}\]
\[x = 2\pi \] satisfies the equation.
Substitute \[x = \dfrac{{5\pi }}{2}\] in the given equation, we get
\[\sin \dfrac{{5\pi }}{4} - \cos \dfrac{{5\pi }}{4} = 1 - \sin \dfrac{{5\pi }}{2}\]
Substituting the values of \[\sin \dfrac{{3\pi }}{2}\] and \[\cos \dfrac{{3\pi }}{2}\], we get
\[\begin{array}{l} \Rightarrow \dfrac{{ - 1}}{{\sqrt 2 }} + \dfrac{1}{{\sqrt 2 }} = 1 - 1\\ \Rightarrow 0 = 0\end{array}\]
\[x = \dfrac{{5\pi }}{2}\] satisfies the equation.
So the values of \[x = \dfrac{\pi }{2},\pi ,2\pi ,\dfrac{{5\pi }}{2}\] .
By comparing the coefficient of \[x\] , we get \[n = 1,2,4,5\]

Therefore, \[n = 1,2,4,5\]

Note:
Note: Here, we need to keep in mind different values of the trigonometric function of \[\sin \theta \] and \[\cos \theta \]. If we do not remember the values we might substitute wrong values in the equation and hence get wrong answers. We might commit a mistake by leaving the solution after solving the inequality and getting the value of \[x\]. This will give us the wrong values of \[n\]. We need to find all values of \[x\] which satisfies the given equation to find the correct values of \[n\].