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Question Number 58521 by malwaan last updated on 24/Apr/19

$$\mathit{s}\mathit{i}\mathit{n}{16}^{°}=?$$$$\mathit{c}\mathit{o}\mathit{s}{16}^{°}=?$$$$\mathit{w}\mathit{i}\mathit{t}\mathit{h}\mathit{o}\mathit{u}\mathit{t}\mathit{u}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathit{c}\mathit{o}\mathit{s}\left(3\theta \right)$$$$\mathit{a}\mathit{l}\mathit{g}\mathit{e}\mathit{b}\mathit{r}\mathit{i}\mathit{c}\mathit{m}\mathit{e}\mathit{t}\mathit{h}\mathit{o}\mathit{d}\mathit{p}\mathit{l}\mathit{e}\mathit{a}\mathit{s}\mathit{e}$$

Answered by tanmay last updated on 24/Apr/19

$$tryingcalculusmethod$$$$y=sinx$$$$\frac{dy}{dx}=cosx$$$$\frac{dy}{dx}\approx \frac{△y}{△x}$$$$△y=\frac{dy}{dx}△x$$$$nowx={15}^{o}x+△x={16}^{o}so△x=1^o=\frac{\pi }{180}radian$$$$sin{15}^o=sin(45-30)=sin{45}^{o}cos{30}^o-cos{45}^{o}sin{30}^o$$$$sin{15}^o=\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}\times \frac{1}{2}=\frac{\sqrt{3}-1}{2\sqrt{2}}$$$$cos{15}^o=cos({45}^o-{30}^o)=\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\times \frac{1}{2}$$$$cos{15}^o=\frac{\sqrt{3}+1}{2\sqrt{2}}$$$$△y=\frac{dy}{dx}\times △x$$$$△y=\left(cosx\right)△x$$$$△y=cos{15}^o\times \frac{\pi }{180}$$$$△y=\left(\frac{\sqrt{3}+1}{2\sqrt{2}}\right)\times \frac{\pi }{180}$$$$hencesin{16}^o=\frac{\sqrt{3}-1}{2\sqrt{2}}+\frac{\sqrt{3}+1}{2\sqrt{2}}\times \frac{\pi }{180}$$$$sin{16}^o=0.2726776311$$$$sin{16}^o\approx 0.2726$$$$nowify=cosx$$$$\frac{dy}{dx}=\frac{△y}{△x}=-sinx$$$$△y=(-sinx)△x$$$$=(-\frac{\sqrt{3}-1}{2\sqrt{2}})\times \frac{\pi }{180}$$$$=(-0.0045172445)$$$$socos\left({16}^o\right)=\frac{\sqrt{3}+1}{2\sqrt{2}}-0.0045172445$$$$cos{16}^o=0.9614085818\approx 0.9614$$

Commented by malwaan last updated on 25/Apr/19

$$thankyousomuchsir$$

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