Question

solve for theta [0,2pi] sin^4(theta)+4cos^2(theta)=1

Answer 1

Hint :\[\sin^4 \theta+\cos^4 \theta=(\sin^2 \theta+\cos^2 \theta)^2-2\sin^2 \theta \cos^2 \theta\]

Answer 2

wait there is a 4 there sorry

Answer 3

maybe write\[\sin^4 \theta=(1-\cos^2 \theta)^2\]

Answer 4

It is obvious that
\[
\theta =\pm \frac\pi 2
\]
are solutions

Answer 5

so pi/2 and 3pi/2

Answer 6

are those the only solutions?

Answer 7

I think there are some more. Stay tuned.

Answer 8

Also we can show that the equation is equivalent to
\[
\cos ^2(\theta ) \left(5 \cos
^2(\theta )+2\right)=0
\]

Answer 9

so, the only answers are pi/2 and 3pi/2

Answer 10

No the other ones are in your interval too.

Answer 11

Oh.

Answer 12

Wait a minute. I am examining something.

Answer 13

Only pi/2 and 3pi/2

Answer 14

thanks. :)

Answer 15

Because \(5 \cos ^2(\theta )+2\ >0\)

Answer 16

yw