Defined Amplitude

Geometry Level 4

Total number of solutions of $[sin x ] + cos x =0$ , where [.] denotes greatest integer function , for $x$ belongs to [0 , 2 $pi$ ]

2 solutions

Hjalmar Orellana Soto
Jul 21, 2015

Ravi Dwivedi
Jul 4, 2015

Since [sinx] is an integer we get three cases using the property of sine function -1<= sinx <=1

Case1: [sinx]=1

In this case sinx =1

So x=pi/2

But cos(pi/2)=0

Putting back in LHS x=pi/2 does not equate to zero

So no solution in this case.

Case2: [sinx]=0

In this case 0<= sinx <1

x belongs to [0,pi/2) U (pi/2,pi]

Our hypothesis in this cases forces cos x=0 which implies x=pi/2

But x cannot be equal to pi/2 because [sinx]=0 by assumption.

Hence no solutions in this case.

Case3: [sinx]=-1

In this case -1< sinx<0

x belongs to (pi,2pi)

For equaton to be true cos x=1 but this is not possible in the interval (pi,2pi)

So no solutions in this case also.

Final answer=0

All of you please guide me how to write solutions. This way I am not able to properly write all the solutions. Also seems ugly.

Thanks!!

Are you sure that $\cos x = 0 \Rightarrow x = \pi /2$ ? Are there no other solutions?

Good casework otherwise.

Sorry for a complete solution its also x=3pi/2 which too does not belong to the required set.

My mistake was due to my focus in the set 0 to pi.

By the way please tell me how to post a solution.

Ravi Dwivedi - 5 years, 11 months ago