Question

What is the value of sin^2(α )+sin^2(β )+sin^2(γ) ? Why?

Answer 1

Answer on Question #55277 – Math – Trigonometry

What is the value of $\sin^2(\alpha) + \sin^2(\beta) + \sin^2(\gamma)$ ? Why?

Solution

The direction angles $\alpha, \beta$ and $\gamma$ are the angles that the vector makes with the positive $x-$ , $y-$ and $z-$ axes, respectively.

If $\alpha, \beta$ and $\gamma$ are the direction angles, then $\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1$ .

Compute

$$\begin{array}{l} \sin^ {2} (\alpha) + \sin^ {2} (\beta) + \sin^ {2} (\gamma) = (1 - \cos^ {2} (\alpha)) + (1 - \cos^ {2} (\beta)) + (1 - \cos^ {2} (\gamma)) = \\ = 3 - \left(\cos^ {2} (\alpha) + \cos^ {2} (\beta) + \cos^ {2} (\gamma)\right) = 3 - 1 = 2 \end{array}$$

If $\alpha, \beta$ and $\gamma$ are the angles of any triangle (where $\alpha + \beta + \gamma = 180{}^\circ$ and each of $\alpha, \beta$ , and $\gamma$ is greater than zero), then $\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) + 2\cos(\alpha)\cos(\beta)\cos(\gamma) = 1$ .

Compute

$$\begin{array}{l} \sin^ {2} (\alpha) + \sin^ {2} (\beta) + \sin^ {2} (\gamma) = (1 - \cos^ {2} (\alpha)) + (1 - \cos^ {2} (\beta)) + (1 - \cos^ {2} (\gamma)) = \\ = 3 - \left(\cos^ {2} (\alpha) + \cos^ {2} (\beta) + \cos^ {2} (\gamma) + 2 \cos (\alpha) \cos (\beta) \cos (\gamma)\right) + 2 \cos (\alpha) \cos (\beta) \cos (\gamma) = \\ = 3 - 1 + 2 \cos (\alpha) \cos (\beta) \cos (\gamma) = 2 + 2 \cos (\alpha) \cos (\beta) \cos (\gamma). \end{array}$$

Answer: $\sin^2 (\alpha) + \sin^2 (\beta) + \sin^2 (\gamma) = 2$

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