The given complex number is
z = 1 + cos(2θ) + i sin(2θ), for -1/2π < θ < 1/2π
Part (i)
Let V = the modulus of z.
Then
V² = [1 + cos(2θ)]² + sin²(2θ)
= 1 + 2 cos(2θ) + cos²2θ + sin²2θ
Because sin²x + cos²x = 1, therefore
V² = 2(1 + cos2θ)
Because cos(2x) = 2 cos²x - 1, therefore
V² = 2(1 + 2cos²θ - 1) = 4 cos²θ
Because -1/2π < θ < 1/2π,
V = 2 cosθ PROVEN
Part ii.
1/z = 1/[1 + cos2θ + i sin 2θ]
The denominator is
Therefore
The real part of 1/ = 1/ (constant).
Answer:
Step-by-step explanation:
Let c represent the number of children that attended the concert.
Let a represent the number of adults that attended the concert.
At one of the concerts, each adult brought 4 children with them, and 10 children attended without an adult. This means that the number of children that attended is
c = 4a + 10
Tickets for a concert were $5 for each child and $8 for each adult. The total ticket sales were $1730. It means that
5c + 8a = 1730
Therefore, the systems of equations that can be solved to determine the number of children, c, and adults that attended the concert is
c = 4a + 10
5c + 8a = 1730
In the general case in Cartesian coordinates, you would use the definition of a parabola as the locus of points equidistant from the focus and directrix. The equation would equate the square of the distance from a general point (x, y) to the focus with the square of the distance from that point to the directrix line.
Suppose the focus is located at (h, k) and the equation of the directrix is ax+by+c=0. The expression for the square of the distance from (x, y) to the point (h, k) is ...
(d₁)² = (x-h)²+(y-k)²
The expression for the square of the distance from (x, y) to the directrix line is
(d₂)² = (ax+by+c)²/(a²+b²)
Equating these expressions gives the equation of the parabola.
(x-h)²+(y-k)² = (ax+by+c)²/(a²+b²)
When the directrix is parallel with one of the axes, one of the coefficents "a" or "b" is zero and the equation becomes much simpler. Often, it would be easier to make use of the formula (for a directrix parallel to the x-axis):
y = 1/(4p)*(x -h)² +k
where the (h, k) here is the vertex, the point halfway between the focus and directrix, and "p" is the (signed) distance from the focus to the vertex. (p is positive when the focus is above or to the right of the vertex.)
21 is the answer multiply time 40 is the real answer
Step-by-step explanation:
There you go
Answer:
The answer is pretty complicated. The percent of obese Americans is over the top. But its really the amount of the people who actually are 18 plus in the District of Columbia.
Step-by-step explanation:
