Question

If \(sinα+sinβ+sinγ=3\), then \(cosα+cosβ+cosγ =\)

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is A

We are given that $\sin \alpha + \sin \beta + \sin \gamma = 3$. We know that the maximum value of the sine function is 1, i.e., $-1 \leq \sin x \leq 1$ for any angle $x$. Therefore, for the sum $\sin \alpha + \sin \beta + \sin \gamma$ to equal 3, each of the terms must equal 1. This means: $$ \sin \alpha = 1 \\ \sin \beta = 1 \\ \sin \gamma = 1 $$ For $\sin \alpha = 1$, we have $\alpha = \frac{\pi}{2} + 2n\pi$ for some integer $n$. For $\sin \beta = 1$, we have $\beta = \frac{\pi}{2} + 2m\pi$ for some integer $m$. For $\sin \gamma = 1$, we have $\gamma = \frac{\pi}{2} + 2p\pi$ for some integer $p$. Therefore, the cosine values are: $$ \cos \alpha = \cos\left(\frac{\pi}{2} + 2n\pi\right) = 0 \\ \cos \beta = \cos\left(\frac{\pi}{2} + 2m\pi\right) = 0 \\ \cos \gamma = \cos\left(\frac{\pi}{2} + 2p\pi\right) = 0 $$ So, $\cos \alpha + \cos \beta + \cos \gamma = 0 + 0 + 0 = 0$.