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Solution 4.3:7a

From Förberedande kurs i matematik 1

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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=1sin2x=1(23)2=35=1sin2y=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.