Solve the system x-y=pi/3 sinx+siny=3^1/2

To solve the system means to find the pair of values for x
and y that verifies the equations of the system.

We'll
re-write the second equation of the system, transforming the sum of like trigonometric
functions into a product:

sin a + sin b = 2sin
[(a+b)/2]*cos [(a-b)/2]

sinx + siny = 2sin [(x+y)/2]*cos
[(x-y)/2]

We'll substitute x - y by pi/3 and we'll
get:

sinx + siny = 2sin [(x+y)/2]*cos
[(pi/3)/2](1)

But sinx + siny = sqrt 3
(2)

We'll put (1) = (2):

2sin
[(x+y)/2]*cos [(pi/3)/2] = sqrt 3

We'll divide by
2:

sin [(x+y)/2]*cos [(pi/6)] = (sqrt
3)/2

We'll substitute cos [(pi/6)] =
sqrt3/2

(sqrt3/2)*sin [(x+y)/2] =
sqrt3/2

We'll divide by
sqrt3/2:

sin [(x+y)/2] =
1

(x+y)/2 = arcsin 1

(x+y)/2 =
pi/2

x + y = 2pi/2

x + y = pi
(3)

x - y = pi/3 (4)

We'll add
(3) + (4):

x + y + x - y = pi +
pi/3

We'll combine and eliminate like
terms:

2x = 4pi/3

We'll divide
by 2:

x =
4pi/6

x =
2pi/3

We'll substitute x in
(3):

2pi/3 + y = pi

We'll
subtract 2pi/3 both sides:

y = pi -
2pi/3 

y =
pi/3

The solution of the
system is represented by the pair (2pi/3 ; pi/3).