Question

How do you find the exact value `sin(x+y)` if `cosx=8/17, siny=12/37`?

Answer 1

The exact value is `=621/629`

Explanation:

We know that

`sin(x+y)=sinxcosy+sinycosx`

If `cosx=8/17` and `siny=12/37`

We can use, `cos^2x+sin^2x=1`

and `cos^2y+sin^2y=1`

To calculate `sinx` and `cosy`

`sin^2x=1-cos^2x=1-(8/17)^2=225/17^2`

`sinx=15/17`

`cos^2y=1-sin^2y=1-(12/37)^2=1225/37^2`

`cosy=35/37`

so,
`sin(x+y)=15/17*35/37+12/37*8/17=621/629`

Answer 2

`sin(x+y)=621/629` or `-429/629` depending on the quadrant in which sine and cosine lie.

Explanation:

Before we commence further, it may be mentioned that as `cosx=8/17`, `x` is in `Q1` or `Q4` i.e. `sinx` could be positive or negative and as `siny=12/37`, `y` is in `Q1` or `Q2` i.e. `cosy` could be positive or negative.

Hence four combinations for `(x+y)` are there and for `sin(x+y)=sinxcosy+cosxsiny`, there are four possibilities.

Now as `cosx=8/17`, `sinx=sqrt(1-(8/17)^2)=sqrt(1-64/289)=sqrt(225/289)=+-15/17` and

as `siny=12/37`, `cosy=sqrt(1-(12/37)^2)=sqrt(1-144/1369)=sqrt(1225/1369)=+-35/37`

Hence,

(1) when `x` and `y` are in `Q1`

`sin(x+y)=15/17xx35/37+8/17xx12/37=(525+96)/629=621/629`

(2) when `x` is in `Q1` and `y` is in `Q2`

`sin(x+y)=15/17xx(-35)/37+8/17xx12/37=(-525+96)/629=-429/629`

(3) when `x` is in `Q4` and `y` is in `Q2`

`sin(x+y)=(-15)/17xx(-35)/37+8/17xx12/37=(525+96)/629=621/629`

(4) when `x` is in `Q4` and `y` is in `Q1`

`sin(x+y)=(-15)/17xx35/37+8/17xx12/37=(-525+96)/629=-429/629`

Hence, `sin(x+y)=621/629` or `-429/629`