Question #194265

A water wheel rotates through the angle x, the water level L behind the wheel changes according the equation "L = 1 - cos x - 2 sin^2 x" where L is measured in inches. Determine all values of x in degrees for which the water level is zero.


1
Expert's answer
2021-05-26T15:56:25-0400

L=1cosx2sin2xL=1−cosx−2sin^2x

PutL=0 in L=1cosx2sin2x,we getPut L=0 \space in \space L=1−cosx−2sin^2x, we \space get

1cosx2sin2x=0⇒1−cosx−2sin^2x=0

1cosx2(1cos2x)=0  [sin2x+cos2x=1]1cosx2+2cos2x=0⇒1−cosx−2(1−cos^2x)=0 \space \space [∵sin^2x+cos^2x=1]\\⇒1−cosx−2+2cos2x=0

2cos2xcosx1=02cos2x2cosx+cosx1=0⇒2cos^2x−cosx−1=0\\⇒2cos^2x−2cosx+cosx−1=0

2cosx(cosx1)+(cosx1)=0(cosx1)(2cosx+1)=0cosx1=0or2cosx+1=0⇒2cosx(cosx−1)+(cosx−1)=0\\⇒(cosx−1)(2cosx+1)=0\\⇒cosx−1=0 or 2cosx+1=0

cosx=1 or 2cosx=1cosx=1 or cosx=1/2x=360norx=120+360n or x=240+360n,nZ⇒cosx=1\space or\space 2cosx=−1\\⇒cosx=1\space or \space cosx=−1/2\\⇒x=360^{\circ}n \,or \\ \,x=120^{\circ}+360^{\circ}n \space or\space x=240^{\circ}+360^{\circ}n, n∈ℤ


Hence the values of x for which the water level is zero are,

x=360n or x=120+360n or x=240+360n,nZx=360^{\circ}n\space or\space x=120^{\circ}+360^{\circ}n\space or \space x=240^{\circ}+360^{\circ}n, n∈ℤ


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!
LATEST TUTORIALS
APPROVED BY CLIENTS