Trigonometry and Polar Coordinates - AP PreCalculus Study Guide

The Core Idea: Trigonometry and Polar Coordinates

The rectangular (or Cartesian) coordinate system describes a point's location using horizontal ($x$) and vertical ($y$) distances from an origin. The polar coordinate system offers an alternative way to describe the same point's location. Instead of using two linear distances, it uses a single directed distance and an angle. A point is defined by its directed distance, $r$, from a central point called the pole, and a directed angle, $\theta$, measured from a fixed ray called the polar axis. This system is fundamentally linked to trigonometry, as the relationships between the sides and angles of a right triangle provide the exact formulas needed to convert between the familiar $(x,y)$ rectangular system and the $(r,\theta )$ polar system.

A key feature of the polar system is that, unlike the rectangular system, each point does not have a unique coordinate pair. A single point can be represented by an infinite number of polar coordinates. This occurs because angles can be expressed as coterminal angles (by adding or subtracting full rotations) and because the directed distance $r$ can be negative, which involves a reflection through the pole. Understanding these multiple representations is as crucial as knowing the conversion formulas themselves.

Key Formulas: Polar-Rectangular Conversion

The relationship between a point's rectangular coordinates $(x,y)$ and its polar coordinates $(r,\theta )$ is defined by a set of four fundamental equations derived from right-triangle trigonometry.

Polar to Rectangular Conversion

These formulas are used when you know $(r,\theta )$ and need to find $(x,y)$.

• $$x=r\cos (\theta )$$

  This formula calculates the horizontal coordinate ($x$) by taking the product of the directed distance from the pole ($r$) and the cosine of the angle ($\theta$).

• $$y=r\sin (\theta )$$

  This formula calculates the vertical coordinate ($y$) by taking the product of the directed distance from the pole ($r$) and the sine of the angle ($\theta$).

Rectangular to Polar Conversion

These formulas are used when you know $(x,y)$ and need to find $(r,\theta )$.

• $$r^2=x^2+y^2$$

  Derived directly from the Pythagorean theorem, this formula relates the polar distance $r$ to the rectangular coordinates. To find $r$, you take the square root: $r=\sqrt{x^2+y^2}$. While $r$ can be negative, this formula typically yields the positive value for $r$.

• $$\tan (\theta )=\frac{y}{x}$$

  This formula relates the polar angle $\theta$ to the rectangular coordinates. To find $\theta$, you would use the arctangent function, $\theta =\arctan \left(\frac{y}{x}\right)$. However, this requires careful consideration of the quadrant in which the point $(x,y)$ lies, as the arctangent function has a limited range.

Understanding Multiple Representations

A defining characteristic of the polar coordinate system is that any single point in the plane can be described by infinitely many different polar coordinate pairs. This is a direct consequence of the periodic nature of trigonometric functions and the concept of a directed distance. The rules for finding these equivalent representations are based on Essential Knowledge statement 3.13A2.

A point with polar coordinates $(r,\theta )$ can also be represented by:

1. Adding Full Rotations:$(r,\theta +2\pi k)$ for any integer $k$.

   • Concept: The terminal side of an angle remains unchanged if you add or subtract full rotations ($2\pi$ radians or $360^{\circ }$).

   • Example: The point $(5,\frac{\pi }{3})$ is the exact same point as $(5,\frac{\pi }{3}+2\pi )=(5,\frac{7\pi }{3})$ (where $k=1$) and $(5,\frac{\pi }{3}-2\pi )=(5,-\frac{5\pi }{3})$ (where $k=-1$). The distance $r$ remains the same, and the angle points in the same direction.

2. Using a Negative Radius:$(-r,\theta +\pi +2\pi k)$ for any integer $k$.

   • Concept: A negative radius, $-r$, means you face in the direction of the angle but move $r$ units in the opposite direction. This is equivalent to adding $\pi$ radians ($180^{\circ }$) to the angle and using a positive radius, $r$.

   • Example: Consider the point $(5,\frac{\pi }{3})$. To represent this with $r=-5$, we must find an angle that points in the opposite direction. The opposite direction is $\frac{\pi }{3}+\pi =\frac{4\pi }{3}$. Therefore, $(5,\frac{\pi }{3})$ is the same point as $(-5,\frac{4\pi }{3})$. We can then add full rotations to this new angle, so $(-5,\frac{4\pi }{3}+2\pi k)$ also works. For instance, $(-5,\frac{4\pi }{3}-2\pi )=(-5,-\frac{2\pi }{3})$ is another valid representation.

Core Concepts & Rules

• Polar Coordinates: A point is located by $(r,\theta )$, where $r$ is the directed distance from the pole (origin) and $\theta$ is the directed angle from the polar axis (positive x-axis).

• Directed Distance ($r$): If $r>0$, the point is $r$ units from the pole along the terminal side of $\theta$. If $r<0$, the point is $|r|$ units from the pole in the direction opposite the terminal side of $\theta$.

• Directed Angle ($\theta$): Positive angles are measured counter-clockwise from the polar axis. Negative angles are measured clockwise.

• Uniqueness: Polar coordinates are not unique. Every point has infinite representations.

• Coterminal Angles: The representation $(r,\theta )$ is equivalent to $(r,\theta +2\pi k)$ for any integer $k$.

• Opposite Direction: The representation $(r,\theta )$ is equivalent to $(-r,\theta +\pi +2\pi k)$ for any integer $k$.

• Conversion to Rectangular: To find $(x,y)$ from $(r,\theta )$, use $x=r\cos (\theta )$ and $y=r\sin (\theta )$.

• Conversion to Polar: To find $(r,\theta )$ from $(x,y)$, use $r=\sqrt{x^2+y^2}$ and $\tan (\theta )=\frac{y}{x}$. Always use the signs of $x$ and $y$ to determine the correct quadrant for $\theta$.

Step-by-Step Example 1: Converting from Polar to Rectangular

Problem: Find the rectangular coordinates $(x,y)$ for the point given by the polar coordinates $(r,\theta )=(-3,\frac{5\pi }{4})$.

Step 1: Identify $r$ and $\theta$ and the conversion formulas.

• The given polar coordinates are $(r,\theta )=(-3,\frac{5\pi }{4})$.

• So, $r=-3$ and $\theta =\frac{5\pi }{4}$.

• The conversion formulas are $x=r\cos (\theta )$ and $y=r\sin (\theta )$.

Step 2: Calculate the $x$-coordinate.

• Substitute the values of $r$ and $\theta$ into the formula for $x$.

  $$x=-3\cos \left(\frac{5\pi }{4}\right)$$

• Evaluate the trigonometric function. The angle $\frac{5\pi }{4}$ is in Quadrant III, where cosine is negative. The reference angle is $\frac{\pi }{4}$, and $\cos (\frac{\pi }{4})=\frac{\sqrt{2}}{2}$.

• Therefore, $\cos (\frac{5\pi }{4})=-\frac{\sqrt{2}}{2}$.

• Complete the calculation for $x$.

  $$x=-3\left(-\frac{\sqrt{2}}{2}\right)=\frac{3\sqrt{2}}{2}$$

Step 3: Calculate the $y$-coordinate.

• Substitute the values of $r$ and $\theta$ into the formula for $y$.

  $$y=-3\sin \left(\frac{5\pi }{4}\right)$$

• Evaluate the trigonometric function. The angle $\frac{5\pi }{4}$ is in Quadrant III, where sine is also negative. $\sin (\frac{\pi }{4})=\frac{\sqrt{2}}{2}$.

• Therefore, $\sin (\frac{5\pi }{4})=-\frac{\sqrt{2}}{2}$.

• Complete the calculation for $y$.

  $$y=-3\left(-\frac{\sqrt{2}}{2}\right)=\frac{3\sqrt{2}}{2}$$

Step 4: State the final answer.

• The rectangular coordinates for the point are $\left(\frac{3\sqrt{2}}{2},\frac{3\sqrt{2}}{2}\right)$.

Step-by-Step Example 2: Exam-Style Application

Problem: The point $P$ has rectangular coordinates $(x,y)=(-5,5\sqrt{3})$. Find a set of polar coordinates $(r,\theta )$ for $P$ that satisfies the conditions $r>0$ and $0\le \theta <2\pi$. Then, find a second representation $(r^{\prime },{\theta }^{\prime })$ that satisfies $r^{\prime }<0$ and $0\le {\theta }^{\prime }<2\pi$.

Part 1: Find the representation with $r>0$.

Step 1: Calculate $r$ using $r^2=x^2+y^2$.

• We are given $x=-5$ and $y=5\sqrt{3}$.

  $$r^2=(-5{)}^2+(5\sqrt{3}{)}^2$$

  $$r^2=25+(25\cdot 3)=25+75=100$$

• Since the condition is $r>0$, we take the positive square root.

  $$r=\sqrt{100}=10$$

Step 2: Find the reference angle for $\theta$ using $\tan (\theta )=\frac{y}{x}$.

$$\tan (\theta )=\frac{5\sqrt{3}}{-5}=-\sqrt{3}$$

• The reference angle $\alpha$ for which $\tan (\alpha )=\sqrt{3}$ is $\alpha =\frac{\pi }{3}$.

Step 3: Determine the correct quadrant and value for $\theta$.

• The point $(x,y)=(-5,5\sqrt{3})$ has a negative $x$-coordinate and a positive $y$-coordinate. This places the point in Quadrant II.

• To find the angle in Quadrant II with a reference angle of $\frac{\pi }{3}$, we calculate $\theta =\pi -\alpha$.

  $$\theta =\pi -\frac{\pi }{3}=\frac{2\pi }{3}$$

• This value satisfies the condition $0\le \theta <2\pi$.

Step 4: State the first polar representation.

• The polar coordinates are $(r,\theta )=(10,\frac{2\pi }{3})$.

Part 2: Find the representation with $r^{\prime }<0$.

Step 5: Determine the new radius $r^{\prime }$.

• The condition is $r^{\prime }<0$. We use the negative of the magnitude we found earlier.

  $$r^{\prime }=-10$$

Step 6: Determine the new angle ${\theta }^{\prime }$ using the rule $(r,\theta )=(-r,\theta +\pi )$.

• We take the angle from Part 1 and add $\pi$ to it.

  $${\theta }^{\prime }=\theta +\pi =\frac{2\pi }{3}+\pi =\frac{2\pi }{3}+\frac{3\pi }{3}=\frac{5\pi }{3}$$

• This angle $\frac{5\pi }{3}$ is in Quadrant IV, which is opposite to Quadrant II. This is the correct geometric relationship.

• This value satisfies the condition $0\le {\theta }^{\prime }<2\pi$.

Step 7: State the second polar representation.

• The second set of polar coordinates is $(r^{\prime },{\theta }^{\prime })=(-10,\frac{5\pi }{3})$.

Using Your Calculator

While many problems on the AP Exam will involve standard angles from the unit circle, a graphing calculator is a powerful tool for converting coordinates with non-standard angles or for checking your work. Ensure your calculator is in Radian mode for most Precalculus problems.

Polar to Rectangular Conversion

To convert $(r,\theta )$ to $(x,y)$`, you can use the built-in functions or the direct formulas.

Problem: Convert the polar coordinates $(r,\theta )=(7,2.5)$ to rectangular coordinates.

Method 1: Direct Formula Entry

1. Calculate $x=r\cos (\theta )$: Type 7 * cos(2.5) and press ENTER.

   • Result: $x\approx -5.609$

2. Calculate $y=r\sin (\theta )$: Type 7 * sin(2.5) and press ENTER.

   • Result: $y\approx 4.187$

3. The rectangular coordinates are approximately $(-5.609,4.187)$.

Method 2: Built-in Functions (TI-84 style)

1. Find the polar-to-rectangular conversion functions, often in the ANGLE menu (accessed via 2nd + APPS).

2. To find $x$, select $P►Rx($. Enter the arguments as $P►Rx(r,\theta )$.

   • Type P►Rx(7, 2.5) and press ENTER. Result: $-5.609$.

3. To find $y$, select $P►Ry($. Enter the arguments as $P►Ry(r,\theta )$.

   • Type P►Ry(7, 2.5) and press ENTER. Result: $4.187$.

Rectangular to Polar Conversion

To convert $(x,y)$ to (r, \theta)Formula[15] such that $r > 0 and $-\pi \le \theta <\pi$."

• Coordinate Identification from a Graph: You might be shown a point plotted in the polar plane and asked to identify its coordinates from a list of options, which may include different valid representations.

  • Example: "A point is plotted at a distance of 4 units from the pole on the ray $\theta =\frac{4\pi }{3}$. Which of the following could be the coordinates of this point? (A) $(4,\frac{\pi }{3})$ (B) $(-4,\frac{\pi }{3})$ (C) $(4,-\frac{\pi }{3})$ (D) $(-4,\frac{4\pi }{3})$"

Common Mistakes

• Quadrant Error in R-to-P Conversion: When converting $(x,y)$ to $(r,\theta )$$, students correctly find ${\theta }_{ref}=\arctan (|\frac{y}{x}|)$ but then fail to place the angle in the correct quadrant based on the signs of $x$ and $y$. For example, for $(-2,-2)$, $\arctan (1)=\frac{\pi }{4}$, but the correct angle is in Quadrant III, $\frac{5\pi }{4}$.

• Misinterpreting Negative $r$: Students often mistake $(-r,\theta )$ for $(r,-\theta )$. A negative $r$ means moving in the direction opposite the terminal ray of $\theta$, which is equivalent to adding $\pi$ to the angle, not negating the angle. For example, $(-2,\frac{\pi }{4})$ is in Quadrant III, not Quadrant IV.

• Incorrectly Applying $\theta +\pi$: When finding an alternate representation with a negative radius, students might correctly negate $r$ but forget to add $\pi$ to $\theta$, or they might add $\pi$ to $\theta$ but forget to negate $r$. Both must be done simultaneously: $(r,\theta )\to (-r,\theta +\pi )$.

• Calculator Mode Error: Performing calculations in Degree mode when the problem is stated in radians (or vice versa). This is a frequent source of incorrect decimal answers on calculator-active questions. Always check your calculator's mode.