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From Förberedande kurs i matematik 1

We can write the expression sin(x+y) in terms of sinx, cosx, siny and cosy if we use the addition formula for sine,

sin(x+y)=sinxcosy+cosxsiny.

In turn, it is possible to express the factors cosx and cosy in terms of sinx and siny by using the Pythagorean identity,

cosxcosy=1−sin2x=1−(23)2=35=1−sin2y=1−(13)2=322.

Because x and y are angles in the first quadrant, cosx and cosy are positive, so we in fact have

cosx=35andcosy=322.

Finally, we obtain

sin(x+y)=32322+3531=942+5.