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Question Number 12132 by tawa last updated on 14/Apr/17

Given that  1° = 0.017 rad  Use  f(a) = sin(a) to find an approximate value for sin(29)°.

$$\mathrm{Given}\:\mathrm{that}\:\:\mathrm{1}°\:=\:\mathrm{0}.\mathrm{017}\:\mathrm{rad} \\ $$$$\mathrm{Use}\:\:\mathrm{f}\left(\mathrm{a}\right)\:=\:\mathrm{sin}\left(\mathrm{a}\right)\:\mathrm{to}\:\mathrm{find}\:\mathrm{an}\:\mathrm{approximate}\:\mathrm{value}\:\mathrm{for}\:\mathrm{sin}\left(\mathrm{29}\right)°. \\ $$

Answered by mrW1 last updated on 14/Apr/17

((Δy)/(Δx))=((f(x+Δx)−f(x))/(Δx))≈(dy/dx)  ⇒ f(x+Δx)≈f(x)+(dy/dx)Δx  y=f(x)=sin x  (dy/dx)=cos x  sin 29°=sin (30−1)≈sin 30°+(dy/dx)×Δx  =(1/2)−((√3)/2)×0.0175=0.4848

$$\frac{\Delta{y}}{\Delta{x}}=\frac{{f}\left({x}+\Delta{x}\right)−{f}\left({x}\right)}{\Delta{x}}\approx\frac{{dy}}{{dx}} \\ $$$$\Rightarrow\:{f}\left({x}+\Delta{x}\right)\approx{f}\left({x}\right)+\frac{{dy}}{{dx}}\Delta{x} \\ $$$${y}={f}\left({x}\right)=\mathrm{sin}\:{x} \\ $$$$\frac{{dy}}{{dx}}=\mathrm{cos}\:{x} \\ $$$$\mathrm{sin}\:\mathrm{29}°=\mathrm{sin}\:\left(\mathrm{30}−\mathrm{1}\right)\approx\mathrm{sin}\:\mathrm{30}°+\frac{{dy}}{{dx}}×\Delta{x} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}×\mathrm{0}.\mathrm{0175}=\mathrm{0}.\mathrm{4848} \\ $$

Commented by tawa last updated on 14/Apr/17

wow, God bless you sir

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

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