Question

The solution set of the system of equation $$x + y = \frac{{2\pi }}{3},\cos x + \cos y = \frac{3}{2},$$       where $$x$$ and $$y$$ are real, is

A. $$x = \frac{\pi }{3} - n\pi ,y = n\pi $$
B. $$\phi $$  
C. $$x = n\pi ,y = \frac{\pi }{3} - n\pi $$
D. None of these
Answer :   $$\phi $$
Solution :
$$\eqalign{ & {\text{We have, }}\cos x + \cos y = \frac{3}{2} \cr & \Rightarrow 2\cos \left( {\frac{{x + y}}{2}} \right)\cos \left( {\frac{{x - y}}{2}} \right) = \frac{3}{2} \cr & \Rightarrow \cos \left( {\frac{{x - y}}{2}} \right) = \frac{3}{2}\,\left( {\because x + y = \frac{{2\pi }}{3}} \right) \cr} $$
Which is not possible $$\left( {{\text{as }}\cos \theta \leqslant 1} \right)$$
Thus, the solution set is a null set.

Releted MCQ Question on
Trigonometry >> Trignometric Equations

Releted Question 1

The equation $$2\,{\cos ^2}\frac{x}{2}{\sin ^2}x = {x^2} + {x^{ - 2}};0 < x \leqslant \frac{\pi }{2}$$        has

A. no real solution
B. one real solution
C. more than one solution
D. none of these
Releted Question 2

The general solution of the trigonometric equation $$\sin x + \cos x = 1$$    is given by:

A. $$x = 2n\pi ;\,\,n = 0,\,\, \pm 1,\,\, \pm 2\,.....$$
B. $$x = 2n\pi + \frac{\pi }{2};\,\,n = 0,\,\, \pm 1,\,\, \pm 2\,.....$$
C. $$x = n\pi + {\left( { - 1} \right)^n}\,\,\frac{\pi }{4} - \frac{\pi }{4}$$
D. none of these
Releted Question 3

The general solution of $$\sin \,x - 3\,\sin \,2x\, + \sin \,3x\, = \cos x - 3\,\cos \,\,2x + \cos \,3x$$           is

A. $$n\pi + \frac{\pi }{8}$$
B. $$\frac{{n\pi }}{2} + \frac{\pi }{8}$$
C. $${\left( { - 1} \right)^n}\frac{{n\pi }}{2} + \frac{\pi }{8}$$
D. $$2n\pi + {\cos ^{ - 1}}\frac{3}{2}$$
Releted Question 4

Number of solutions of the equation $$\tan x + \sec x = 2\cos x$$     lying in the interval $$\left[ {0,2\pi } \right]$$  is:

A. 0
B. 1
C. 2
D. 3

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Trignometric Equations


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