Question

If 180° < α < 270°, cos⁡ α = −817, 270° < β < 360°, and sin⁡ β = −45, what is cos⁡ (α + β)?

Answer 1

`cos(\alpha+\beta)=-(84)/(85)`

Explanation:

we know that

`cos(\alpha+\beta)=cos(\alpha)*cos(\beta)-sin(\alpha)*sin(\beta)`

Remember the identity

`cos^(2) (x)+sin^2(x)=1`

step 1

Find the value of
`sin(\alpha)`

we have that

The angle alpha lie on the III Quadrant

so

The values of sine and cosine are negative

`cos(\alpha)=-(8)/(17)`

Find the value of sine

`cos^(2) (\alpha)+sin^2(\alpha)=1`

substitute

`(-(8)/(17))^(2)+sin^2(\alpha)=1`

`sin^2(\alpha)=1-(64)/(289)`

`sin^2(\alpha)=(225)/(289)`

`sin(\alpha)=-(15)/(17)`

step 2

Find the value of
`cos(\beta)`

we have that

The angle beta lie on the IV Quadrant

so

The value of the cosine is positive and the value of the sine is negative

`sin(\beta)=-(4)/(5)`

Find the value of cosine

`cos^(2) (\beta)+sin^2(\beta)=1`

substitute

`(-(4)/(5))^(2)+cos^2(\beta)=1`

`cos^2(\beta)=1-(16)/(25)`

`cos^2(\beta)=(9)/(25)`

`cos(\beta)=(3)/(5)`

step 3

Find cos⁡ (α + β)

`cos(\alpha+\beta)=cos(\alpha)*cos(\beta)-sin(\alpha)*sin(\beta)`

we have

`cos(\alpha)=-(8)/(17)`

`sin(\alpha)=-(15)/(17)`

`sin(\beta)=-(4)/(5)`

`cos(\beta)=(3)/(5)`

substitute

`cos(\alpha+\beta)=-(8)/(17)*(3)/(5)-(-(15)/(17))*(-(4)/(5))`

`cos(\alpha+\beta)=-(24)/(85)-(60)/(85)`

`cos(\alpha+\beta)=-(84)/(85)`