Solve cos(4x)-cos(2x)=0 ∀ 0<=x<=2pi ..............(0)
Normal solution:
1. use the double angle formula to decompose, and recall cos^2(x)+sin^2(x)=1
cos(4x)=cos^2(2x)-sin^2(2x)=2cos^2(2x)-1 .................(1)
2. substitute (1) in (0)
2cos^2(2x)-1-cos(2x)=0
3. substitute u=cos(2x)
2u^2-u-1=0
4. Solve for x
factor
(u-1)(u+1/2)=0
=> u=1 or u=-1/2
However, since cos(x) is an even function, so solutions to
{cos(2x)=1, cos(-2x)=1, cos(2x)=-1/2 and cos(-2x)} ...........(2)
are all solutions.
5. The cosine function is symmetrical about pi, therefore
cos(-2x)=cos(2*pi-2x),
solution (2) above becomes
{cos(2x)=1, cos(2pi-2x)=1, cos(2x)=-1/2, cos(2pi-2x)=-1/2}
6. Solve each case
cos(2x)=1 => x=0
cos(2pi-2x)=1 => cos(2pi-0)=1 => x=pi
cos(2x)=-1/2 => 2x=2pi/3 or 2x=4pi/3 => x=pi/3 or 2pi/3
cos(2pi-2x)=-1/2 => 2pi-2x=2pi/3 or 2pi-2x=4pi/3 => x=2pi/3 or x=4pi/3
Summing up,
x={0,pi/3, 2pi/3, pi, 4pi/3}
The answer is 2x+3 <span>Twice a certain number is tripled. The resulting number is _____.
2x + 3
2x - 3
(2x)3
1/3(2x)</span>
Answer:
solution: x= 0.273, y= 1.909.
Step-by-step explanation:
- given system of linear equations : x+3y=64x−6y=6
- these can also be splitted or written individually as:
x+3y=6 & 64x−6y=6
- the first equation can also be written as,
so, it is line with slope : -1/3 and y-intercept :2
- the second equation can also be written as,
so, it is line with slope : 32/3 and y-intercept : -1
- now we can easily draw graphs for these equations using the above interpretation.
- graph is in the attachment.
- so, the solution is: x= 0.273, y= 1.909.
Answer:
-4
Step-by-step explanation:
in 6x3-4x4+22 4 is the biggest exponent
the coefficient of the biggest exponent is -4 becuase it would be -4(4)
therefore the leading coefficient is -4
