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Question Number 36739 by tawa tawa last updated on 04/Jun/18

If   x = a(1 − cosθ)i  +  asinθ j  find the resultant of x in its simplest form.

$$\mathrm{If}\:\:\:\mathrm{x}\:=\:\mathrm{a}\left(\mathrm{1}\:−\:\mathrm{cos}\theta\right)\mathrm{i}\:\:+\:\:\mathrm{asin}\theta\:\mathrm{j} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{resultant}\:\mathrm{of}\:\mathrm{x}\:\mathrm{in}\:\mathrm{its}\:\mathrm{simplest}\:\mathrm{form}. \\ $$

Commented by prof Abdo imad last updated on 05/Jun/18

x = 2a sin^2 ((θ/2))i  +2asin((θ/2))cos((θ/2))j  =2a sin((θ/2)){ sin((θ/2))i +cos((θ/2))j} ⇒  ∣∣x∣∣ =2∣asin((θ/2))∣ .

$${x}\:=\:\mathrm{2}{a}\:{sin}^{\mathrm{2}} \left(\frac{\theta}{\mathrm{2}}\right){i}\:\:+\mathrm{2}{asin}\left(\frac{\theta}{\mathrm{2}}\right){cos}\left(\frac{\theta}{\mathrm{2}}\right){j} \\ $$$$=\mathrm{2}{a}\:{sin}\left(\frac{\theta}{\mathrm{2}}\right)\left\{\:{sin}\left(\frac{\theta}{\mathrm{2}}\right){i}\:+{cos}\left(\frac{\theta}{\mathrm{2}}\right){j}\right\}\:\Rightarrow \\ $$$$\mid\mid{x}\mid\mid\:=\mathrm{2}\mid{asin}\left(\frac{\theta}{\mathrm{2}}\right)\mid\:. \\ $$

Commented by tawa tawa last updated on 07/Jun/18

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Commented by maxmathsup by imad last updated on 07/Jun/18

and you too sir.

$${and}\:{you}\:{too}\:{sir}. \\ $$

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