alpha, beta, gamma are real numbers satisfyin | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $\alpha + \beta + \gamma = \pi$. For a triangle with angles $\alpha, \beta, \gamma$,the sum of sines is given by $\sin \alpha + \sin \beta +

Vedclass · Accepted Answer

Positive

Vedclass · Answer

(C) Given $\alpha + \beta + \gamma = \pi$. For a triangle with angles $\alpha, \beta, \gamma$,the sum of sines is given by $\sin \alpha + \sin \beta + \sin \gamma = 4 \cos(\frac{\alpha}{2}) \cos(\frac{\beta}{2}) \cos(\frac{\gamma}{2})$. Since $\alpha, \beta, \gamma$ are angles of a triangle,each angle must be in the interval $(0, \pi)$,which implies $\frac{\alpha}{2}, \frac{\beta}{2}, \frac{\gamma}{2} \in (0, \frac{\pi}{2})$. In the interval $(0, \frac{\pi}{2})$,the cosine function is always positive. Therefore,$4 \cos(\frac{\alpha}{2}) \cos(\frac{\beta}{2}) \cos(\frac{\gamma}{2}) > 0$. Thus,the expression $\sin \alpha + \sin \beta + \sin \gamma$ is always positive.

$\alpha, \beta, \gamma$ are real numbers satisfying $\alpha + \beta + \gamma = \pi$. The minimum value of the expression $\sin \alpha + \sin \beta + \sin \gamma$ is

Similar Questions

If $A + B + C = \pi ,$ then $\cos \, 2A + \cos \, 2B + \cos \, 2C = $

If $\alpha+\beta=\frac{\pi}{2}$ and $\beta+\gamma=\alpha$,then $\tan \alpha$ equals

Given $a^2 + 2a + \csc^2 \left( \frac{\pi}{2}(a + x) \right) = 0$,then which of the following holds good?

The maximum value of $\sin x - \cos x$ is equal to

If the equation $\sin^4 x + \cos^4 x = a$ has real solutions,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module