Solve the equation cos3x=cos2x for 0¡Üx¡Ü360
cos3x = cos2x
we need 3(x+2m*pi) = 2(x+2n*pi) for some m and n. Some playing around will get you
x = 0,2/5 pi, 4/5 pi
or, in degrees,
x = 0,144,288
?? but the ans is 0, 72 ,144,216,288,360 degree
In terms of cosx
cos 3x = 4 cos^3 x - 3cosx , and cos 2x = 2 cos^2 x - 1
so cos 3x = cos 2x
4 cos^3 x - 3cosx - 2cos^2 x + 1 = 0
let cosx = y
so we need
4y^3 - 2y^2 - 3y + 1 = 0
I tried y = 1 and sure enough that was a solution.
then by synthetic division ...
4y^3 - 2y^2 - 3y + 1 = 0
(y-1)((4y^2 + 2y - 1) = 0
y = 1 or y = .30902 or y = -.80902 , using the formula
if y = 1, cosx = 1
x = 0 , 360
if y = .30902 , cosx = .30902
x = 72°, 288
if y = -.80902 , cosx = -.80902
x = 144° , 216°
x = 0, 72, 144, 216, 288, and 360°
Good catch, Reiny - I was a bit sloppy.
To solve the equation cos3x = cos2x, we can use the trigonometric identity:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
In this case, we can rewrite the equation as:
cos(3x - 2x) = cos(2x)cos(x) + sin(2x)sin(x)
Which simplifies to:
cos(x) = cos(2x)cos(x) + sin(2x)sin(x)
Now, we can subtract cos(2x)cos(x) from both sides:
cos(x) - cos(2x)cos(x) = sin(2x)sin(x)
Factoring out cos(x) on the left-hand side:
cos(x)(1 - cos(2x)) = sin(2x)sin(x)
Now, we can divide both sides by sin(2x):
cos(x)(1 - cos(2x))/sin(2x) = sin(x)
We know that cos(x)/sin(x) is the same as cot(x), so we can rewrite the equation as:
cot(x)(1 - cos(2x))/2 = 1
Multiplying both sides by 2, we get:
cot(x)(1 - cos(2x)) = 2
Expanding and rearranging the equation:
cot(x) - cot(x)cos(2x) = 2
Now, we can use the identity cot(A) = cos(A)/sin(A):
cos(x)/sin(x) - cos(x)(1 - 2sin^2(x)) = 2
Next, we need to simplify further by multiplying through by sin(x):
cos(x) - cos(x)(1 - 2sin^2(x))sin(x) = 2sin(x)
Expanding and simplifying:
cos(x) - cos(x)sin(x) + 2sin^3(x)cos(x) = 2sin(x)
We can factor out cos(x) from the first two terms:
cos(x)(1 - sin(x)) + 2sin^3(x)cos(x) = 2sin(x)
Now, we can divide both sides by cos(x):
1 - sin(x) + 2sin^3(x) = 2sin(x)/cos(x)
Since sin(x)/cos(x) is equal to tan(x), we can rewrite the equation as:
1 - sin(x) + 2sin^3(x) = 2tan(x)
Now, we have an equation involving only sin(x) and tan(x). We can solve this equation numerically using an approximation method or by using a graphing calculator/software.