Question

the number of real solutions of the equation (cos x)^7+(sin x)^4=1 in the interval [-π,π] are? and what are they

Answer 1

I can see -pi/2,0,pi/2---> 3 solutions

Answer 2

yup its correct but how did u got those

Answer 3

no but what about other angles in between -pi/2 to pi/2

Answer 4

i know that zero -pi/2 and pi/2 are the solutions but how to confirm it is there any method that other angles wont satisfy

Answer 5

cos^7 +(sin^2)^2 =1
cos^7 +(1-cos^2)^2 =1
cos^7 +1-2cos+cos^2 =1
cos^7+cos^2 -2cos =0
cos(cos^6+cos -2)=0
--> cos x =0 --> x = \(\pm pi/2\)

Answer 6

for cos^6 +cos -2 =0 hehehehe.... I don't know how to solve. by inspection, we can see only x = 0 in the interval give cos x = 1 to get the expression true. but how to put it in neat??? let think more

Answer 7

@phi

Answer 8

well i too was stuck here