waigaK

2021-08-17

Two direction angles of a vector are given. Find the third direction angle, given that is either acute.

$\beta =\frac{2\pi}{3},\gamma =\frac{\pi}{4},\alpha \text{}\text{is acute}$

timbalemX

Skilled2021-08-18Added 108 answers

According to given details

$\beta =\frac{2\pi}{3},\gamma =\frac{\pi}{4}$

Using trigonometry identities

${\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\beta +{\mathrm{cos}}^{2}\gamma =1$

${\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\frac{2\pi}{3}+{\mathrm{cos}}^{2}\frac{\pi}{4}=1$

${\mathrm{cos}}^{2}\alpha +{(-\frac{1}{2})}^{2}+\left(\frac{1}{\sqrt{2}}\right)=$

${\mathrm{cos}}^{2}\alpha \frac{1}{4}+\frac{1}{2}=1$

$\mathrm{cos}}^{2}\alpha =\frac{1}{4$

$\mathrm{cos}\alpha =\frac{1}{2}$

$\alpha =\frac{\pi}{3}$

$\alpha ={60}^{\circ}$

Using trigonometry identities

Jeffrey Jordon

Expert2021-12-10Added 2605 answers

Answer is given below (on video)

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