Use trigonometric identities to calcuate sin(345)

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We have to find sin 345.

sin 345 = sin ( 360 - 15) = - sin 15

Now cos 30 = (cos 15)^2 - ( sin 15)^2

=> cos 30 = 1 - 2*(sin 15)^2

=> 1 - 2*(sin 15)^2 = cos 30

=> - 2*(sin 15)^2 = cos 30 - 1

=> 2 (sin 15)^2 = 1 - cos 30

=> 2 (sin 15)^2 = 1 - sqrt 3/2

=>  (sin 15)^2 = 1/2 - sqrt 3/4

=> (sin 15) = sqrt (1/2 - sqrt 3/4)

=> sin 15 = sqrt (1/2 - sqrt 3/4)

Therefore sin 345 = - (sqrt (1/2 - sqrt 3/4))

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We need to use the trigonometric identities to find the values of sin(345).

First we need to rewrite 345 as a product of two common angles.

==> 345 = 120 + 225

We know that 120 = (60 degrees in the second quadrant)

Also we know that 225 = (45 degrees into the 3rd quadrant).

==> sin345= sin(120+225)

We know that:

sin(a+b) = sina*cosb + cosa*sinb.

==> sin(120+225) = sin120*cos225+sin225*cos120.

                            = sqrt3/2 * -1/sqrt2 + -1/sqrt2* -1/2

                        = -sqrt3/2sqrt2 + 1/2sqrt2

                         = (1-sqrt3)/2sqrt2

==> sin(345) = (1-sqrt3)/2sqrt2

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