Given sinx=1/2+cosx, what is the value of sin2x?

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We know that sin 2x = 2*sin x * cos x

It is given that sin x = 1/2 + cosx

=> sin x - cos x = 1/2

square both the sides

=> (sin x)^2 +(cos x)^2 - 2*sin x * cos x = 1/4

we know that (sin x)^2 +(cos x)^2 = 1

=> 1 - 2*sin x * cos x = 1/4

=> 2*sin x * cos x = 3/4

=> sin 2x = 3/4

The value of sin 2x = 3/4

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll work on the relation given by enunciation:

sinx = 1/2 + cosx

We'll subtract cos x:

sin x - cos x = 1/2

We'll raise to square both sides:

(sin x - cos x)^2 = 1/4

We'll expand the binomial:

(sin x)^2 - 2sin x*cos x + (cos x)^2 = 1/4

We'll apply Pythagorean identity:

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

We'll also apply the double angle identity:

sin 2x = 2sin x*cos x

The relation will become:

1 - sin 2x = 1/4

sin 2x = 1 - 1/4

sin 2x = 3/4

The requested value of sin 2x is: sin 2x = 3/4.

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