It looks like the given equation is
sin(2x) - sin(2x) cos(2x) = sin(4x)
Recall the double angle identity for sine:
sin(2x) = 2 sin(x) cos(x)
which lets us rewrite the equation as
sin(2x) - sin(2x) cos(2x) = 2 sin(2x) cos(2x)
Move everything over to one side and factorize:
sin(2x) - sin(2x) cos(2x) - 2 sin(2x) cos(2x) = 0
sin(2x) - 3 sin(2x) cos(2x) = 0
sin(2x) (1 - 3 cos(2x)) = 0
Then we have two families of solutions,
sin(2x) = 0 or 1 - 3 cos(2x) = 0
sin(2x) = 0 or cos(2x) = 1/3
[2x = arcsin(0) + 2nπ or 2x = π - arcsin(0) + 2nπ]
… … … or [2x = arccos(1/3) + 2nπ or 2x = -arccos(1/3) + 2nπ]
(where n is any integer)
[2x = 2nπ or 2x = π + 2nπ]
… … … or [2x = arccos(1/3) + 2nπ or 2x = -arccos(1/3) + 2nπ]
[x = nπ or x = π/2 + nπ]
… … … or [x = 1/2 arccos(1/3) + nπ or x = -1/2 arccos(1/3) + nπ]
Answer:
1) A point P is named or labeled by using a coordinate pair, where the regular notation is:
P = (x, y).
Where x and y are the horizontal and vertical components.
2) Three distinct points are collinear if exists a line that passes through the 3 points.
3) Three distinct points are complanar if they are not collinear, this means that:
We can have two colinear points (so one line connects them) and another point outside that line.
Then the set of 3 points is coplanar.
8+9+10 = 27
27/3 = 9
4+5+6 = 15
15/3 = 5
This is always so, because the third number has one extra that can be added to the first to make them 3x the same numbers.
The common denominator of these fractions will be 12. So Ann worked 306/12 hours, Mary worked 244/12 hours, and John worked 123 hours, meaning they have worked 673/12 hours together.