Question

If 180°<α<270°, cos ⁡α= −8/17, 270°<β<360°, and sin β= −4/5, what is sin⁡(α+β)?

Answer 1

`\sin(\alpha+\beta) = -0.153`

Explanation:

Let determine the angles behind each trigonometric expression:

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

`\alpha = \cos^(-1)\left(-(8)/(17) \right)`

Given that
`180^(\circ)< \alpha < 270^(\circ)`, the value of
`\alpha` is:

`\alpha \approx 241.928^(\circ)`

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

`\beta = \sin^(-1)\left(-(4)/(5) \right)`

Given that
`270^(\circ)< \beta <360^(\circ)`, the value of
`\beta` is:

`\beta \approx 306.870^(\circ)`

The sine function of the sum of angles can be determined by the following identity:

`\sin(\alpha + \beta)=\sin \alpha \cdot \cos \beta + \sin \beta \cdot \cos \alpha`

If
`\alpha \approx 241.928^(\circ)` and
`\beta \approx 306.870^(\circ)`, then:

`\sin (241.928^(\circ)+306.870^(\circ)) = (\sin 241.928^(\circ)) \cdot (\cos 306.870^(\circ)) + (\sin 306.870^(\circ))\cdot (\cos 241.928^(\circ))`
`\sin(241.928^(\circ)+306.870^(\circ)) = -0.153`