Convert Points Between Polar and Rectangular Coordinates: Videos & Practice Problems

To convert polar coordinates (r, θ) to rectangular coordinates (x, y), use the equations x = r × cos(θ) and y = r × sin(θ). For example, for polar coordinates (5, π/3), x = 5 × cos(π/3) = 5/2 and y = 5 × sin(π/3) = 5√3/2. Conversely, to convert rectangular coordinates (x, y) to polar coordinates, calculate r using r = √(x² + y²) and θ using θ = tan⁻¹(y/x), adjusting for the quadrant. This process emphasizes the relationship between these coordinate systems through trigonometric functions and the Pythagorean theorem.

Convert Points from Polar to Rectangular

Understanding the conversion between polar and rectangular coordinates is essential in mathematics, particularly in trigonometry and geometry. Polar coordinates are expressed as \( (r, \theta) \), where \( r \) is the distance from the origin and \( \theta \) is the angle from the positive x-axis. To convert these coordinates into rectangular coordinates \( (x, y) \), we can utilize the relationships derived from right triangles.

For a point given in polar coordinates, the conversion formulas are:

\[x = r \cdot \cos(\theta)\]\[y = r \cdot \sin(\theta)\]

To illustrate this, consider the polar coordinate \( (5, \frac{\pi}{3}) \). Here, \( r = 5 \) and \( \theta = \frac{\pi}{3} \). By forming a right triangle, we can identify the hypotenuse as \( r \) and the angles as \( \theta \). Using the cosine and sine functions, we can find \( x \) and \( y \):

For \( x \):

\[x = 5 \cdot \cos\left(\frac{\pi}{3}\right) = 5 \cdot \frac{1}{2} = \frac{5}{2}\]

For \( y \):

\[y = 5 \cdot \sin\left(\frac{\pi}{3}\right) = 5 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}\]

Thus, the rectangular coordinates corresponding to the polar point \( (5, \frac{\pi}{3}) \) are \( \left(\frac{5}{2}, \frac{5\sqrt{3}}{2}\right) \).

Next, consider the polar coordinate \( (-3, \frac{\pi}{6}) \). The negative \( r \) value indicates that we will move in the opposite direction of the angle. First, we plot the angle \( \frac{\pi}{6} \) and then move 3 units in the opposite direction. The conversion to rectangular coordinates follows the same formulas:

For \( x \):

\[x = -3 \cdot \cos\left(\frac{\pi}{6}\right) = -3 \cdot \frac{\sqrt{3}}{2} = -\frac{3\sqrt{3}}{2}\]

For \( y \):

\[y = -3 \cdot \sin\left(\frac{\pi}{6}\right) = -3 \cdot \frac{1}{2} = -\frac{3}{2}\]

Thus, the rectangular coordinates for \( (-3, \frac{\pi}{6}) \) are \( \left(-\frac{3\sqrt{3}}{2}, -\frac{3}{2}\right) \).

Lastly, when the polar coordinate is \( (0, -\frac{\pi}{6}) \), the \( r \) value of 0 indicates that the point is at the origin. Regardless of the angle, the rectangular coordinates will be:

\[x = 0 \cdot \cos\left(-\frac{\pi}{6}\right) = 0\]

\[y = 0 \cdot \sin\left(-\frac{\pi}{6}\right) = 0\]

Therefore, the rectangular coordinates are \( (0, 0) \), which corresponds to the origin in both coordinate systems.

In summary, converting between polar and rectangular coordinates involves using trigonometric functions to derive the \( x \) and \( y \) values based on the angle and radius. This skill is fundamental for graphing and understanding the relationships between different coordinate systems.

Convert the point to rectangular coordinates.
$(-2,-\frac{\pi}{4})$

• A. $(\sqrt{2},-\sqrt2)$
• B. $(-1,1)$
• C. $(-\sqrt{2},\sqrt{2})$
• D. $(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$

Answer: (C)

Convert the point to rectangular coordinates.
$(4,\frac{\pi}{6})$

• A. $(2\sqrt{3},2)$
• B. $(4\sqrt{3},4)$
• C. $(2,2\sqrt{3})$
• D. $(2,\sqrt{3})$

Answer: (A)

Convert the point to rectangular coordinates.
$(-3,0)$

• A. $(-3,0)$
• B. $(0,-3)$
• C. $(0,0)$
• D. $(3,0)$

Answer: (A)

Convert the point to rectangular coordinates.
$(0,\frac{7\pi}{4})$

• A. $(0,\frac74)$
• B. $(-4,0)$
• C. $(0,0)$
• D. $(0,7)$

Answer: (C)

Convert Points from Rectangular to Polar

To convert points from rectangular coordinates to polar coordinates, we utilize the relationships between the two systems, which are fundamentally based on right triangles. Given a point in rectangular coordinates, represented as (x, y), we can derive the polar coordinates (r, θ) using the Pythagorean theorem and trigonometric functions.

First, we calculate the radial distance r, which is the hypotenuse of the triangle formed by the x and y coordinates. The formula for r is:

$$ r = \sqrt{x^2 + y^2} $$

For example, for the point (3, 4):

$$ r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

Next, we determine the angle θ using the tangent function, which relates the opposite side (y) to the adjacent side (x):

$$ \tan(θ) = \frac{y}{x} $$

For the same point (3, 4):

$$ θ = \tan^{-1}\left(\frac{4}{3}\right) \approx 53^\circ $$

Thus, the polar coordinates for the point (3, 4) are (5, 53°).

When dealing with points in different quadrants, it is crucial to consider the location of the point to determine the correct angle. For instance, for the point (-4, 0), we find:

$$ r = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4 $$

Since this point lies on the negative x-axis, the angle θ is π radians (or 180°), giving us polar coordinates (4, π).

For a point like (-1, √3), we first calculate:

$$ r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 $$

Next, we find θ:

$$ θ = \tan^{-1}\left(\frac{\sqrt{3}}{-1}\right) $$

This yields a negative angle, indicating it is in quadrant IV. To adjust for quadrant II, we add π:

$$ θ = -\frac{\pi}{3} + \pi = \frac{2\pi}{3} $$

Thus, the polar coordinates for the point (-1, √3) are (2, 2π/3).

In summary, the process for converting rectangular coordinates to polar coordinates involves plotting the point, calculating r using the Pythagorean theorem, and determining θ using the tangent function while considering the quadrant in which the point lies. This systematic approach ensures accurate conversion across all quadrants.

Convert Points from Rectangular to Polar Example 1

To convert the point (3, -3) from rectangular coordinates to polar coordinates, we follow a systematic approach. First, we plot the point on the rectangular coordinate system, which is located in the fourth quadrant.

Next, we calculate the radial distance, denoted as r, using the formula:

r = \sqrt{x^2 + y^2}

Substituting the values of x and y (3 and -3, respectively), we find:

r = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}

Now, we determine the angle θ. Since the point is in the fourth quadrant, we use the inverse tangent function:

θ = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{-3}{3}\right) = \tan^{-1}(-1)

This simplifies to:

θ = -\frac{\pi}{4}

It is important to note that angles in polar coordinates can be represented in multiple ways. For instance, the angle -π/4 can also be expressed as 7π/4, which is equivalent in the context of polar coordinates. Thus, the polar coordinates for the point (3, -3) can be represented as:

(r, θ) = (3\sqrt{2}, \frac{7\pi}{4})

Both representations are valid and will return to the original rectangular coordinates when converted back. This illustrates the flexibility in expressing points in polar coordinates.

Convert the point to polar coordinates.
$(0,5)$

• A. $(0,0)$
• B. $(5,0)$
• C. $(-5,\frac{\pi}{2})$
• D. $(5,\frac{\pi}{2})$

Answer: (D)

Convert the point to polar coordinates.
$(-2,2)$

• A. $(2\sqrt{2},\frac{3\pi}{4})$
• B. $(2\sqrt{2},-\frac{\pi}{4})$
• C. $(2\sqrt{2},\frac{\pi}{4})$
• D. $(-2\sqrt{2},\frac{3\pi}{4})$

Answer: (A)

Convert the point to polar coordinates.
$(1,1)$

• A. $(1,\frac{\pi}{4})$
• B. $(\sqrt{2},\frac{\pi}{4})$
• C. $(\sqrt{2},\frac{5\pi}{4})$
• D. $(\sqrt{2},-\frac{\pi}{4})$

Answer: (B)

Convert the point to polar coordinates.
$(-1,-\sqrt{3})$

• A. $(2,\frac{\pi}{3})$
• B. $(2,\frac{7\pi}{6})$
• C. $(2,\frac{4\pi}{3})$
• D. $(-2,\frac{4\pi}{3})$

Answer: (C)

Here’s what students ask on this topic:

How do you convert polar coordinates to rectangular coordinates?

To convert polar coordinates (r, θ) to rectangular coordinates (x, y), use the following equations:

$x=r\times \cos \left(\theta \right)$

$y=r\times \sin \left(\theta \right)$

For example, if you have polar coordinates (5, π/3), you can find the rectangular coordinates as follows:

$x=5\times \cos \left(\frac{\pi }{3}\right)=\frac{5}{2}$

$y=5\times \sin \left(\frac{\pi }{3}\right)=\frac{5}{\sqrt{3}}$

How do you convert rectangular coordinates to polar coordinates?

To convert rectangular coordinates (x, y) to polar coordinates (r, θ), follow these steps:

1. Calculate r using the Pythagorean theorem:

$r=\sqrt{x^2+y^2}$

2. Calculate θ using the tangent function:

$\theta =\tan ⁻¹\left(\frac{y}{x}\right)$

Adjust θ based on the quadrant where the point (x, y) is located. For example, for the point (3, 4):

$r=\sqrt{3{+}^4}=5$

$\theta =\tan ⁻¹\left(\frac{4}{3}\right)\approx 53°$

What are the equations for converting polar coordinates to rectangular coordinates?

The equations for converting polar coordinates (r, θ) to rectangular coordinates (x, y) are:

$x=r\times \cos \left(\theta \right)$

$y=r\times \sin \left(\theta \right)$

These equations use the trigonometric functions cosine and sine to find the x and y coordinates from the given r and θ values.

What are the equations for converting rectangular coordinates to polar coordinates?

The equations for converting rectangular coordinates (x, y) to polar coordinates (r, θ) are:

$r=\sqrt{x^2+y^2}$

$\theta =\tan ⁻¹\left(\frac{y}{x}\right)$

These equations use the Pythagorean theorem to find r and the inverse tangent function to find θ. Adjust θ based on the quadrant where the point (x, y) is located.

How do you find the angle θ when converting from rectangular to polar coordinates?

To find the angle θ when converting from rectangular coordinates (x, y) to polar coordinates, use the equation:

$\theta =\tan ⁻¹\left(\frac{y}{x}\right)$

After calculating θ, adjust it based on the quadrant where the point (x, y) is located:

• Quadrant I: θ is as calculated.
• Quadrant II: Add π to θ.
• Quadrant III: Add π to θ.
• Quadrant IV: Add 2π to θ if θ is negative.

This ensures θ is correctly positioned in the polar coordinate system.