Answer to Question #139926 in Trigonometry for Michael

Question #139926
Marion needs to find the cosine of Pi over 12. If she knows that cos Pi over 6 = root 3 over 2, how can she use this fact to find the cosine of Pi over 12? What is her answer. Please explain and show your work
1
Expert's answer
2020-10-26T19:19:29-0400

Use double angle formula for cosine cos(2A)=2cos2(A)1cos(2A)=2cos^2(A)-1


So, the cosine of angle AA can be written as,


2cos2(A)=1+cos(2A)2cos^2(A)=1+cos(2A)


cos2(A)=12(1+cos(2A))cos^2(A)=\frac{1}{2}(1+cos(2A))


cos(A)=12(1+cos(2A))cos(A)=\sqrt{\frac{1}{2}(1+cos(2A))}


Plug A=π12A=\frac{\pi}{12} into the relation cos(A)=12(1+cos(2A))cos(A)=\sqrt{\frac{1}{2}(1+cos(2A))} to obtain,


cos(π12)=12(1+cos(2(π12)))cos(\frac{\pi}{12})=\sqrt{\frac{1}{2}(1+cos(2(\frac{\pi}{12})))}


=12(1+cos(π6))=\sqrt{\frac{1}{2}(1+cos(\frac{\pi}{6}))}


=12(1+32)=\sqrt{\frac{1}{2}(1+\frac{\sqrt{3}}{2})} .....plug cos(π6)=32cos(\frac{\pi}{6})=\frac{\sqrt3}{2}


=12(2+32)=\sqrt{\frac{1}{2}(\frac{2+\sqrt{3}}{2})}


=14(2+3)=\sqrt{\frac{1}{4}(2+\sqrt{3})}


=122+30.965926=\frac{1}{2}\sqrt{2+\sqrt{3}}\approx0.965926


Therefore, the cosine of π12\frac{\pi}{12} is cos(π12)=122+30.965926cos(\frac{\pi}{12})=\frac{1}{2}\sqrt{2+\sqrt{3}}\approx0.965926

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment