The rectangular and polar coordinates of a point are (x, y) and (r, θ), where x= 4 and θ= 46◦.
a. Determine the value of r.
b. Determine the value of y.
x = r cos T
y = r sin T
4 = r cos 46
r = 4/cos 46 = 5.76
y = 5.76 sin 46 = 4.14
a. To find the value of r, we can use the formula:
r = √(x^2 + y^2)
Given that x = 4, we can substitute this value into the formula:
r = √(4^2 + y^2)
Since this is a quadratic equation, we need more information to determine the value of y.
b. To find the value of y, we can use the formula:
y = r * sin(θ)
Given that θ = 46°, and we are trying to find y, we can set up the equation:
y = r * sin(46°)
However, since we don't know the value of r, we cannot determine the exact value of y.
To determine the value of r, we need to use the conversion formulas between rectangular and polar coordinates:
For a point (x, y) in rectangular coordinates:
r = √(x^2 + y^2)
For a point (r, θ) in polar coordinates:
x = r * cos(θ)
y = r * sin(θ)
Given that x = 4 and θ = 46 degrees (∠ is the angle sign), we can use the equation x = r * cos(θ) to find r.
a. Determine the value of r:
x = r * cos(θ)
4 = r * cos(46°)
To isolate r, divide both sides by cos(46°):
r = 4 / cos(46°)
Using a calculator, compute the value of cos(46°) and then divide 4 by that value to find r.
b. Determine the value of y:
We can use the equation y = r * sin(θ) to find the value of y.
Substitute the known values:
y = r * sin(θ)
= (4 / cos(46°)) * sin(46°)
Using a calculator, compute the value of cos(46°) and sin(46°), and then multiply (4 / cos(46°)) by sin(46°) to find the value of y.