For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve.
(a) A line through the origin that makes an angle of π/3 with the positive x-axis.
(b) A vertical line through the point (4, 4).

a) is the equation of the curve that makes an angle π/3.

b) is the equation of line through the point (4,4).

Step-by-step explanation:

Given:

A line from origin which makes an angle of with x-axis.

A vertical line from .

We have to write the equation of the curves in Polar or Cartesian format.

Step wise:

a) A line from origin which makes an angle of with x-axis.

To write the equation of the above line in Polar coordinates is more desirable as the angles could be defined well in polar form.

So,

⇒ ...equation (i)

⇒ ...here is the slope

The slope in terms of (angle) can be written as,

⇒

Plugging the values of the angle, .

⇒ ...equation (ii)

Now re-arranging the equation (i) we can write it as,

⇒  

b) A vertical line from .

Note:

The equation of a vertical line always takes the form x = k, where k is any number and k is also the x-intercept .

To write the above point in Cartesian coordinate is more acceptable and easy for us.

⇒

Then,

y = sq-rt(3) x is the equation of the curve that makes an angle π/3.

and x = 3 is the equation of line through the point (4,4).

the expression (1.26)2t reveals the approximate rate of increases in the number of plants semiannually.

hope this : )

for the friend request.

when a figure, such as triangle (xyz) is reflected over the line y = x to create x'y'z' the x-coordinates and y-coordinates of the original image (pre-image) will be reversed for the image. (e.g., (x, y) > (y, x)) this means that the pre-image position of the x- and y-coordinates switch places which in turn provides a mirror like image of the original xyz.  

i notice that when i draw a line segment from x to x', that the points are parallel from each other -- they are exactly across from each other and do not intersect with any other points.  

yes, think i would see the same characteristic if you drew the line segment connecting y with the reflecting line and then y' with the reflecting line because of the essence of reflections. because it is a mirror image, each point was effected the same way which leads me to conclude that they same would apply to y and y'.