Verified answer

sin 555° = sin (360° + 195°) = sin 195°

since sin (x+y) = sin x( cos y) + sin y( cos x)

sin (360 + 195) = sin 360(cos 195) + sin 195(cos 360) = sin 195 Since sin 360° =0 and cos 360° = 1

sin 195° = sin (180 + 15°) = sin 180°(cos 15°) + cos 180°(sin 15°) = -sin 15°

-sin 15° = -sin (45° - 30°) = -sin 45°(cos 30°) -(-)sin 30°(cos 45°) = - (√2/2)(√3/2) +(1/2)(√2/2)

-sin 15° = - √6/4 +√2/4 = (√2 -√6)/4

So sin 555° = (√2 - √6)/4

First, since 555 is greater than 360 degrees, let's get the reference angle

555 - 360 =>

195

Since sin(555) = sin(195), this makes it a whole lot easier to work with

195 = 225 - 30

sin(195) =>

sin(225 - 30) =>

sin(225)cos(30) - sin(30)cos(225) =>

(-sqrt(2)/2) * (sqrt(3)/2) - (1/2) * (-sqrt(2)/2) =>

(sqrt(2)/4) * (-sqrt(3) + 1) =>

(sqrt(2) / 4) * (1 - sqrt(3))

sin(555°) = sin(555°-360°)

= sin(195°)

= sin(180°+15°)

= -sin(15°)

= -sin(30°/2)

= -√((1-cos 30°)/2)

= -√((1-√3/2)/2)

= -√((2-√3)/4)

= -(1/2)√(2-√3)).

In these calculations, I used the identities

sin(x°-360°) = sin(x°) (line 1)

sin(180°+x°) = -sin(x°) (line 4)

sin(x°/2) = √((1-cos(x°))/2) when 0º≤xº≤180º (line 6)

I would take two steps. First, sin(555) = sin(360 + 195)=sin(360)cos(195)+cos(360)sin(195)=sin(195).

Now take sin(195)=sin(135+60) = sin(135)cos(60) + cos(135)sin(60) = (sqrt(2)/2)(1/2) +

(-sqrt(2)/2)(sqrt(3)/2) = sqrt(2)/4 - sqrt(6)/4.

sin 15 = sin (40 5 - 30) distinction formula sin (A - B) = sin A cos B - cos A sin B sin 15 = sin 40 5 cos 30 - cos 40 5 sin 30 sin 15 = ((sqrt 2)/2)((sqrt 3)/2) - ((sqrt 2)/2) (a million/2) sin 15 = (sqrt 6)/4 - (sqrt 2)/4 sin 15 = (sqrt 6 - sqrt 2)/4