A community for students.
Here's the question you clicked on:
 0 viewing
mukushla
 3 years ago
Solve this equation\[\sin^6 x+\cos^6 x=\frac{5}{8}\]for \(x \in[0,2\pi]\)
mukushla
 3 years ago
Solve this equation\[\sin^6 x+\cos^6 x=\frac{5}{8}\]for \(x \in[0,2\pi]\)

This Question is Closed

sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.2sin^6 x + cos^6 x (sin^2x)^3+(cos^2x)^3 (sin^2x + cos^2x) (sin^4xsin^2x cos^2x +cos^4x) 1*{(sin^2x +cos^2x)^2 3sin^2x cos^2x} 13sin^2xcos^2x 13(sinx cosx)^2 13(sin2x/2)^2 13/4 sin^2(2x)

sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.213/4 sin^2 (2x) =5/8 3/4 sin^2 (2x)=3/8 sin^2 (2x)=1/2 2sin^2 (2x) =1 12sin^2 (2x)=0 cos4x=0

sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.2Now LET a=4x then, cos(a)=0 , 0<=a<=8pi

sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.2I hope this will lead to the solution.

sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.2And remember x=a/4

sauravshakya
 3 years ago
Best ResponseYou've already chosen the best response.2Is that correct @mukushla

harsh314
 3 years ago
Best ResponseYou've already chosen the best response.0i think if you do it like this it would be shorter \[ (1\cos^{2}x)^{ 3}\]\[(1\cos^{2}x)^{ 3} +\cos ^{6}x=\frac{ 5}{ 8 }\] \[1\cos ^{6}x3\cos ^{2}x(1\cos ^{2}x)+\cos ^{6}x=\frac{ 5 }{ 8 }\] the two cos^6 terms cancel \[3\cos ^{2}x \times \sin ^{2}x=\frac{ 5 }{ 8 }1\] \[3\sin ^{2}x \times \cos ^{2}x=\frac{ 3 }{ 8 }\] multiply both sides by 4/3 and we have\[4\sin ^{2}xcos ^{2}x=\frac{ 1 }{ 2 }\] and 2sinx cosx=sin2x so \[\sin ^{2}2x=\frac{ 1 }{ 2 }\] hence \[2x=\frac{ \pi }{ 4 } or 2x=\frac{ 3\pi }{ 4 }\] hence x is pi/8 or 3pi/8

harsh314
 3 years ago
Best ResponseYou've already chosen the best response.0it is looking cumbersome but if you work it out it wont be so.........
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.