Question

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

Answer 1

Answer:

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

Step-by-step explanation:

Let determine the angles behind each trigonometric expression:

$\cos \alpha = -\frac{8}{17}$

$\alpha = \cos^{-1}\left(-\frac{8}{17} \right)$

Given that $180^{\circ}< \alpha < 270^{\circ}$, the value of $\alpha$ is:

$\alpha \approx 241.928^{\circ}$

$\sin \beta = -\frac{4}{5}$

$\beta = \sin^{-1}\left(-\frac{4}{5} \right)$

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$