What is the value of sum of sqrt2/2+sinx=?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

This is a weird question, it is not possible to determine "the value of sum of sqrt2/2+sinx=?" as sin x is a variable.

I guess you want the value of x for which sqrt2/2+sinx= 0

(sqrt 2)/2 + sin x = 0

=> sin x = -(sqrt 2)/2

=> sin x = -1/(sqrt 2)

x = arc sin (-1/(sqrt 2))

=> x = 225 degrees

and x = 315 degrees.

Also, sine is a periodic function.

The solution of the equation: sqrt2/2+sinx= 0 is x = 225 + n*360 degrees and 315 + n*360 degrees.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To determine the value of the sum, we'll create matching functions in the given sum.

Since sin pi/4 = (sqrt2)/2, we'll substitute the value (sqrt2)/2 by the equivalent function of the angle pi/4.

We'll transform the sum  into a product.

sin x + sin pi/4 = sin x + (sqrt2)/2

sin x + sin pi/4  =  2sin [(x+pi/4)/2]*cos[ (x-pi/4)/2]

sin x + sin pi/4  = 2 sin [(x/2 + pi/8)]*cos[ (x/2 - pi/8)]

We'll use the half angle identity:

sin [(x+pi/4)/2] = sqrt[2-(sqrt2)*(cos x-sin x)]/2

cos[ (x-pi/4)/2] = sqrt[2+(sqrt2)*(cos x+sin x)]/2

sin x + sin pi/4  = sqrt{[2-(sqrt2)*(cos x-sin x)]*[2+(sqrt2)*(cos x+sin x)]}/2

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