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Question Number 208282 by hardmath last updated on 10/Jun/24

$$\mathrm{If}\cos \alpha -\cos \beta =\frac{1}{5}\sin \alpha +\sin \beta =\frac{1}{2}$$$$\mathrm{Find}\cos (\alpha +\beta )=?$$

Commented by mr W last updated on 12/Jun/24

$$whatyouwrotemeans$$$$\cos \alpha -\cos \beta =\frac{1}{2}$$$$\frac{1}{5}\sin \alpha +\sin \beta =\frac{1}{2}$$$$butactuallyyoumeant$$$$\cos \alpha -\cos \beta =\frac{1}{5}$$$$\sin \alpha +\sin \beta =\frac{1}{2}$$$$theyareverydifferentquestions!$$$${it}^{\prime }smisleading!$$$$canyoupleasepaymoreattention?$$

Commented by Frix last updated on 10/Jun/24

$$\mathrm{I}\mathrm{solved}\mathrm{both}\mathrm{versions}\mathrm{but}\mathrm{one}\mathrm{gives}\mathrm{no}$$$$\mathrm{useable}\mathrm{exact}\mathrm{result},\mathrm{thus}\mathrm{I}\mathrm{think}\mathrm{the}$$$$\mathrm{other}\mathrm{one}\mathrm{was}\mathrm{meant}.$$

Answered by Frix last updated on 10/Jun/24

$$\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$$$${(\cos \alpha -\cos \beta =\frac{1}{5})}^2\Rightarrow$$$$\cos \alpha \cos \beta =\frac{{\cos }^2\alpha +{\cos }^2\beta }{2}-\frac{1}{50}$$$${(\sin \alpha +\sin \beta =\frac{1}{2})}^2\Rightarrow$$$$\sin \alpha \sin \beta =\frac{1}{8}-\frac{{\sin }^2\alpha +{\sin }^2\beta }{2}$$$$\Rightarrow$$$$\cos (\alpha +\beta )=\frac{171}{200}$$

Commented by efronzo1 last updated on 10/Jun/24

$$\cos \alpha -\cos \beta =\frac{1}{2}$$

Commented by hardmath last updated on 10/Jun/24

$$\mathrm{yes}\mathrm{dear}\mathrm{professor}\mathrm{thank}\mathrm{you}$$

Commented by hardmath last updated on 10/Jun/24

$$\mathrm{answer}:171/200$$

Commented by efronzo1 last updated on 10/Jun/24

$$\underbrace{}$$

Commented by Frix last updated on 10/Jun/24

$$\mathrm{Then}\mathrm{solve}\mathrm{it}\mathrm{if}\mathrm{you}\mathrm{can}!$$

Commented by efronzo1 last updated on 11/Jun/24

$$\mathrm{read}\mathrm{the}\mathrm{original}\mathrm{question}$$$$\cos \alpha -\cos \beta =\frac{1}{5}\sin \alpha +\sin \beta =\frac{1}{2}$$$$\mathrm{but}\mathrm{your}\cos \alpha -\cos \beta =\frac{1}{5}$$$$\mathrm{and}\sin \alpha +\sin \beta =\frac{1}{2}$$

Commented by mr W last updated on 11/Jun/24

$$theconfusionwascausedby$$$$\mathit{h}\mathit{a}\mathit{r}\mathit{d}\mathit{m}\mathit{a}\mathit{t}\mathit{h}sirthroughhis$$$$misleadingwritingofthequestion.$$

Commented by Frix last updated on 11/Jun/24

@efronzo1: As I wrote before I also solved this one but it gives no useable exact result, we can only approximate it. So I decided that the other version with the "nice" result was the right one.

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