How do you convert the polar coordinate ( 4.5541 , 1.2352 ) into cartesian coordinates?

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How do you convert the polar coordinate ( 4.5541 , 1.2352 ) into cartesian coordinates?

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Answer 1 · 2016-01-29T04:26:23

Polar coordinates are in the form $(r, θ)$ $(r, theta)$ , Cartesian (also called rectangular) coordinates are in the form $(x, y)$ $(x, y)$ . In this case the Cartesian coordinates are $(1.50, 4.30)$ $(1.50, 4.30)$ .

Explanation:

Polar coordinates are stated in the form $(r, θ)$ $(r, theta)$ , a distance from the origin and an angle, in radians, counterclockwise from the positive $x$ $x$ axis. In this case, we have a point $4.5541$ $4.5541$ units from the origin, at an angle of $1.2352$ $1.2352$ $r a d$ $rad$ .

Cartesian coordinates, sometimes also called 'rectangular coordinates', are expressed in the form $(x, y)$ $(x, y)$ where $x$ $x$ is the distance along the $x$ $x$ axis and $y$ $y$ is the distance up the $y$ $y$ axis.

To convert from one to the other, we use trigonometry. The $r$ $r$ value from the polar coordinates is the hypotenuse of a right-angled triangle, and the $x$ $x$ and $y$ $y$ coordinates are the adjacent and opposite sides respectively, from the perspective of the angle $θ$ $theta$ .

Knowing the angle and a side allows us to use the definitions of sine and cosine to find the $x$ $x$ and $y$ $y$ coordinates:

$sin θ = \frac{o p p o s i t e}{hypotenuse} \to$ $sin theta = (opposite)/("hypotenuse") to$ $o p p o s i t e = hypotenuse \cdot sin θ$ $opposite = "hypotenuse"* sin theta$

In this case, $y = 4.5541 \cdot sin (1.2352) = 4.5541 \cdot 0.9442 = 4.30$ $y = 4.5541*sin(1.2352) = 4.5541*0.9442=4.30$

(Be sure to ensure that your calculator is on the 'radians', not 'degrees' setting when calculating the sine and cosine in this context.)

By similar reasoning:

$cos θ = \frac{a d j a c e n t}{hypotenuse} \to$ $cos theta = (adjacent)/("hypotenuse") to$ $a d j a c e n t = hypotenuse \cdot cos θ$ $adjacent = "hypotenuse"* cos theta$

In this case, $x = 4.5541 \cdot cos (1.2352) = 4.5541 \cdot 0.3293 = 1.50$ $x = 4.5541*cos(1.2352) = 4.5541*0.3293=1.50$

So that you don't have to remember all this discussion every time, you can memorize (but should understand):

$x = r cos θ$ $x = r cos theta$

$y = r sin θ$ $y=rsintheta$

Combine these, and the Cartesian coordinates for the point are $(1.50, 4.30)$ $(1.50, 4.30)$ .

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How do you convert the polar coordinate ( 4.5541 , 1.2352 ) into cartesian coordinates?

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