Kyran Hudson

2021-08-21

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

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

Brighton

Skilled2021-08-22Added 103 answers

According to given details

$\alpha =\frac{\pi}{3},\gamma =\frac{2\pi}{3}$

Using trigonometry identities

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

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

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

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

$\mathrm{cos}}^{2}\beta =1-\frac{1}{2$

$\mathrm{cos}}^{2}\beta =\frac{1}{2$

$\mathrm{cos}\beta =\sqrt{\frac{1}{2}}$

$\beta =\frac{\pi}{4}$

$\beta ={45}^{\circ}$

Using trigonometry identities

Jeffrey Jordon

Expert2021-12-10Added 2605 answers

Answer is given below (on video)

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