Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 101522 by bramlex last updated on 03/Jul/20

calculate : cos^2 47^o +cos^2 73^o +cos 47^o .cos 73^o +(1/2)

$$\mathrm{calculate}\::\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{47}^{\mathrm{o}} +\mathrm{cos}\:^{\mathrm{2}} \mathrm{73}^{\mathrm{o}} +\mathrm{cos}\:\mathrm{47}^{\mathrm{o}} .\mathrm{cos}\:\mathrm{73}^{\mathrm{o}} +\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Commented by bemath last updated on 03/Jul/20

= (cos 47^o +cos 73^o )^2 −cos 73^o .cos 43^o +(1/2)  = (2cos 60^o .cos 13^o )^2 −(1/2)(cos 120^o +cos 26^o )+(1/2)  = cos^2 13^o +(1/4)−(1/2)cos 26^o +(1/2)  =(3/4)+cos^2 13^o −(1/2)(2cos^2 13^o −1)  = (3/4)+(1/2) = (5/4)

$$=\:\left(\mathrm{cos}\:\mathrm{47}^{\mathrm{o}} +\mathrm{cos}\:\mathrm{73}^{\mathrm{o}} \right)^{\mathrm{2}} −\mathrm{cos}\:\mathrm{73}^{\mathrm{o}} .\mathrm{cos}\:\mathrm{43}^{\mathrm{o}} +\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$=\:\left(\mathrm{2cos}\:\mathrm{60}^{\mathrm{o}} .\mathrm{cos}\:\mathrm{13}^{\mathrm{o}} \right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\:\mathrm{120}^{\mathrm{o}} +\mathrm{cos}\:\mathrm{26}^{\mathrm{o}} \right)+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$=\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{13}^{\mathrm{o}} +\frac{\mathrm{1}}{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\mathrm{26}^{\mathrm{o}} +\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$=\frac{\mathrm{3}}{\mathrm{4}}+\mathrm{cos}\:^{\mathrm{2}} \mathrm{13}^{\mathrm{o}} −\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{2cos}\:^{\mathrm{2}} \mathrm{13}^{\mathrm{o}} −\mathrm{1}\right) \\ $$$$=\:\frac{\mathrm{3}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}}\:=\:\frac{\mathrm{5}}{\mathrm{4}} \\ $$

Answered by Dwaipayan Shikari last updated on 03/Jul/20

(cos47°+cos73°)^2 −(1/2)(cos120°+cos26°)+(1/2)  =cos^2 13°−(1/2)(2cos^2 13°−1)+(1/4)+(1/2)  =1+(1/4)=1.25

$$\left({cos}\mathrm{47}°+{cos}\mathrm{73}°\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\left({cos}\mathrm{120}°+{cos}\mathrm{26}°\right)+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$={cos}^{\mathrm{2}} \mathrm{13}°−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{2}{cos}^{\mathrm{2}} \mathrm{13}°−\mathrm{1}\right)+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$=\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}}=\mathrm{1}.\mathrm{25} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com