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calculate the sum sin x+square root(1-sin^2x)

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justaguide eNotes educator | Certified Educator

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We have to calculate sin x + sqrt (  1 - (sin x)^2)

sin x + sqrt (  1 - (sin x)^2)

=> sin x + sqrt ((cos x)^2)

=> sin x + cos x

This sum can be changed to many other forms but the basic result is sin x + cos x

Therefore the sum = sin x + cos x

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khizirsiddiqui | Student

We have the fundamental Identity: cos^2(x) + sin^2(x) = 1
We can manipulate it in the given way -
cos^2(x) = 1 - sin^2(x)
=> cos(x) = ± √( 1 - sin^2(x))

so, we can write sin x + sqrt(1-sin^2x) as
= sin(x) ± cos(x)

The sign of ± cos(x) depends upon the value of x.

giorgiana1976 | Student

To calculate the expression we'll have to transform the sum into a product. The terms of the su are not like trigonometric functions. 

We'll re-write the second term: sqrt(1-sin^2x) = sqrt (cos x)^2

sin x + sqrt(1-sin^2x) = sin x + sqrt (cos x)^2

sin x + sqrt(1-sin^2x) = sin x + cos x

We'll  express the function cosine, depending on the function sine.

cos x= sin (90-x)

The expression will become:

 sin x + cos x = sinx + sin (90-x)

Now we can transform the expression into a product:

 sin x + cos x = 2 sin (x+90-x)/2*cos (x-90+x)/2

 sin x + cos x = 2 sin 45*cos [-(90-2x)/2]

 sin x + cos x =  2* (sqrt2/2)*cos (45-x)

 sin x + cos x =  sqrt 2*(cos 45*cos x + sin 45*sin x)

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