If-b-cos-120-0-c-cos-240-0-then-prove-that-b-c-b-c-3-tan-

Question Number 220855 by fantastic last updated on 20/May/25

If b cos(θ+120^0 )=c cos (θ+240^0 ) then prove that b−c=−(b+c)(√3) tan θ

$${If}\:\:{b}\:\mathrm{cos}\left(\theta+\mathrm{120}^{\mathrm{0}} \right)={c}\:\mathrm{cos}\:\left(\theta+\mathrm{240}^{\mathrm{0}} \right)\:{then}\:{prove}\:{that} \\ $$$${b}−{c}=−\left({b}+{c}\right)\sqrt{\mathrm{3}}\:\mathrm{tan}\:\theta \\ $$

Answered by golsendro last updated on 20/May/25

b cos (180−(60−θ)) = c cos (360−(120−θ)) −b cos (60−θ) = c cos (120−θ) − b cos (60−θ) = c cos (180−(60+θ)) b cos (60−θ) = c cos (60+θ) b((1/2) cos θ+(1/2)(√3) sin θ) = c((1/2) cos θ−(1/2)(√3) sin θ) (b−c) cos θ = −(b+c)(√3) sin θ ⇔ b−c = −(b+c)(√3) tan θ

$$\:\mathrm{b}\:\mathrm{cos}\:\left(\mathrm{180}−\left(\mathrm{60}−\theta\right)\right)\:=\:\mathrm{c}\:\mathrm{cos}\:\left(\mathrm{360}−\left(\mathrm{120}−\theta\right)\right) \\ $$$$\:−\mathrm{b}\:\mathrm{cos}\:\left(\mathrm{60}−\theta\right)\:=\:\mathrm{c}\:\mathrm{cos}\:\left(\mathrm{120}−\theta\right) \\ $$$$\:−\:\mathrm{b}\:\mathrm{cos}\:\left(\mathrm{60}−\theta\right)\:=\:\mathrm{c}\:\mathrm{cos}\:\left(\mathrm{180}−\left(\mathrm{60}+\theta\right)\right) \\ $$$$\:\:\mathrm{b}\:\mathrm{cos}\:\left(\mathrm{60}−\theta\right)\:=\:\mathrm{c}\:\mathrm{cos}\:\left(\mathrm{60}+\theta\right) \\ $$$$\:\mathrm{b}\left(\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{cos}\:\theta+\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{3}}\:\mathrm{sin}\:\theta\right)\:=\:\mathrm{c}\left(\frac{\mathrm{1}}{\mathrm{2}}\:\mathrm{cos}\:\theta−\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{3}}\:\mathrm{sin}\:\theta\right) \\ $$$$\:\left(\mathrm{b}−\mathrm{c}\right)\:\mathrm{cos}\:\theta\:=\:−\left(\mathrm{b}+\mathrm{c}\right)\sqrt{\mathrm{3}}\:\mathrm{sin}\:\theta \\ $$$$\:\Leftrightarrow\:\mathrm{b}−\mathrm{c}\:=\:−\left(\mathrm{b}+\mathrm{c}\right)\sqrt{\mathrm{3}}\:\mathrm{tan}\:\theta \\ $$

Facebook Tweet Pin

If-b-cos-120-0-c-cos-240-0-then-prove-that-b-c-b-c-3-tan-

Leave a Reply Cancel reply