12.

The polar coordinates system is an alternate system to normal (x,y) system. The generic form is `(r, theta )` , where r is the length (usually only the magnitude) to the point from the origin (0,0) and `theta` is counter clockwise angle between the line connecting the point and origin and x-axis. For example `(2, pi/2)` would represent a point on the y axis, 2 lengths above from the origin on the positive side. Sometimes (-r, theta) can be used. This usually means the length is on the opposite side or we can convert it to normal notation using,

`(-r,theta) = (r, theta+-pi)` If we add or subtract 180 or `pi `, the direction inverts.

(i) `(3, 30^0) `

This represents a point from a distance of 3 and angle of `30^0` degrees with the positive direction of x-axis. We can convert these coordinates to (x,y) coordinates using the following relationships.

`x = 2 cos 30^0 and y = 2 sin 30^0`

`x = 2 xx 0.8660 and y = 2 xx 0.5`

`x = 1.732 and y = 1.`

(ii) `(-2, 180^0)`

We can convert the negative length to positive length by,

`(-2, 180^0) = (2, 180^0-180^0)` `= (2, 0^0)`

This is a point within a distance of 2 and on the x axis (as the angle is 0 degrees).

Therefore ,` x = 2, y = 0.`

13.

We can use the "Law of Sines" to solve this problem.

If `a, b, and c` are sides of an triangle and `alpha, beta and gamma` are opposite angles to respective sides in the given order, the "Law of Sine" says,

`a/(sin alpha) = b/(sin beta) = c/(sin gamma) = constant`

In this given triangle, `a, b and alpha` are given.

`a = 1, b=1.8 and alpha = 26^0`

Therefore,

`1/(sin 26) = 1.8/(sin beta)`

`sin beta = (1.8 xx sin 26)/1`

`sin beta = 0.7891`

`beta = sin^-1(0.7891) = 52.1^0`

In a triangle, we know all the internal angles add up to 180 degrees.

Therefore, `alpha +beta+ gamma = 180^0`

`26^0+52.1^0+gamma = 180^0`

`gamma = 101.9^0`

Again, using the "Law of Sines",

`a/(sin alpha) = c/(sin gamma)`

`1/(sin 26) = c/(sin 101.9)`

`c = (1 xx sin 101.9)/(sin 26) = 2.23 `

14.

The point (-3,3) is on the second quarter of the Cartesian coordinates.

The length or radius r is

`r = sqrt((-3)^2+3^2)`

`r = 3 sqrt (2)`

`r = 4.24`

The angle is `theta.`

If the angle with y-axis is `beta,` then,

`tan beta = 3/3 = 1`

`beta =tan^-1(1)`

`beta = 45`

But we need the angle with the x-axis, which is `theta` .

`theta = 90 + beta = 90 + 45 = 135`

Therefore the polar coordinates are,

`(3sqrt(2), 135^0) or (4.24, 135^0)`