Center
The unit circle is centered at the origin, \((0,0)\).
Trigonometry Guide
Unit Circle Explained Step-by-Step
Learn how degrees, radians, coordinates, reference angles, and exact trigonometric values fit together on the unit circle.
The unit circle connects angles with coordinates and trigonometric values. Once you understand this connection, you do not have to treat sine, cosine, and tangent as unrelated formulas to memorize.
In this guide, you will learn how the unit circle is constructed, how degrees and radians describe the same angles, and how each point on the circle gives you exact sine, cosine, and tangent values.
Key Idea
For an angle \(\theta\), the point on the unit circle has coordinates \((\cos\theta,\sin\theta)\).
What Is the Unit Circle?
The unit circle is a circle with a radius of \(1\) centered at the origin of the coordinate plane.
Equation of the Unit Circle
\[
x^2+y^2=1
\]
The center of the circle is \((0,0)\), and every point on the circle is exactly \(1\) unit away from the origin.
Visual Model
A Circle With Radius 1
Center: \((0,0)\)
The rightmost point is \((1,0)\), the top point is \((0,1)\), the leftmost point is \((-1,0)\), and the bottom point is \((0,-1)\).
Radius
The radius always has a length of exactly \(1\).
Points
Every point \((x,y)\) on the circle satisfies \(x^2+y^2=1\).
Quick Check
Is \(\left(\frac{3}{5},\frac{4}{5}\right)\) on the unit circle?
Substitute the coordinates into the equation \(x^2+y^2=1\).
\[
\begin{aligned}
x^2+y^2
&=
\left(\frac{3}{5}\right)^2+
\left(\frac{4}{5}\right)^2 \\[4pt]
&=
\frac{9}{25}+\frac{16}{25} \\[4pt]
&=
\frac{25}{25} \\[4pt]
&=1
\end{aligned}
\]
Answer:
Yes. The point
\(\left(\frac{3}{5},\frac{4}{5}\right)\)
is on the unit circle because its
coordinates satisfy \(x^2+y^2=1\).
Why This Matters
When a radius rotates through an angle \(\theta\), its endpoint lands at a point \((x,y)\) on the unit circle. Those coordinates become the cosine and sine of the angle.
Coordinates and Trigonometric Values
Start at the positive \(x\)-axis and rotate a radius through an angle \(\theta\). The endpoint of that radius lands at a point \((x,y)\) on the unit circle.
Fundamental Unit Circle Relationship
\[
(x,y)
=
(\cos\theta,\sin\theta)
\]
This means that the horizontal coordinate gives the cosine of the angle, while the vertical coordinate gives the sine.
Visual Model
Connecting an Angle to a Point
\(P=(\cos\theta,\sin\theta)\)
The horizontal distance from the origin is \(\cos\theta\), and the vertical distance is \(\sin\theta\).
\(x\)
Cosine
The \(x\)-coordinate of the point is the cosine of the angle.
\[
x=\cos\theta
\]
\(y\)
Sine
The \(y\)-coordinate of the point is the sine of the angle.
\[
y=\sin\theta
\]
\(\dfrac{y}{x}\)
Tangent
Tangent is the ratio of the \(y\)-coordinate to the \(x\)-coordinate.
\[
\tan\theta
=
\frac{\sin\theta}{\cos\theta}
=
\frac{y}{x}
\]
Example 1
Find the trigonometric values from a point on the unit circle
Suppose the terminal side of \(\theta\) intersects the unit circle at the point \[ P= \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right). \]
Since \((x,y)=(\cos\theta,\sin\theta)\), read cosine and sine directly from the coordinates.
\[
\begin{aligned}
\cos\theta
&=
\frac{\sqrt{3}}{2}
\\[6pt]
\sin\theta
&=
\frac{1}{2}
\end{aligned}
\]
To find tangent, divide sine by cosine.
\[
\begin{aligned}
\tan\theta
&=
\frac{\sin\theta}{\cos\theta}
\\[6pt]
&=
\frac{\frac{1}{2}}
{\frac{\sqrt{3}}{2}}
\\[6pt]
&=
\frac{1}{\sqrt{3}}
\\[6pt]
&=
\frac{\sqrt{3}}{3}
\end{aligned}
\]
Answer:
\[
\sin\theta=\frac{1}{2},
\qquad
\cos\theta=\frac{\sqrt{3}}{2},
\qquad
\tan\theta=\frac{\sqrt{3}}{3}
\]
Keep the Coordinate Order Straight
The unit-circle point is always written as \((\cos\theta,\sin\theta)\). Cosine comes first because it is the \(x\)-coordinate. Sine comes second because it is the \(y\)-coordinate.
Degrees and Radians
Degrees and radians are two different units used to measure angles. They describe the same rotations using different numbers.
One Complete Rotation
\[
360^\circ=2\pi\text{ radians}
\]
\[
180^\circ=\pi\text{ radians}
\]
The relationship \(180^\circ=\pi\) radians gives us the conversion factors used to move between the two units.
Degrees to Radians
Multiply by \(\dfrac{\pi}{180^\circ}\)
\[
\text{Degrees}
\times
\frac{\pi}{180^\circ}
=
\text{Radians}
\]
Radians to Degrees
Multiply by \(\dfrac{180^\circ}{\pi}\)
\[
\text{Radians}
\times
\frac{180^\circ}{\pi}
=
\text{Degrees}
\]
Common Degree and Radian Equivalents
| Degrees | Radians | Fraction of a Rotation |
|---|---|---|
| \(0^\circ\) | \(0\) | Start |
| \(30^\circ\) | \(\dfrac{\pi}{6}\) | \(\dfrac{1}{12}\) |
| \(45^\circ\) | \(\dfrac{\pi}{4}\) | \(\dfrac{1}{8}\) |
| \(60^\circ\) | \(\dfrac{\pi}{3}\) | \(\dfrac{1}{6}\) |
| \(90^\circ\) | \(\dfrac{\pi}{2}\) | \(\dfrac{1}{4}\) |
| \(180^\circ\) | \(\pi\) | \(\dfrac{1}{2}\) |
| \(270^\circ\) | \(\dfrac{3\pi}{2}\) | \(\dfrac{3}{4}\) |
| \(360^\circ\) | \(2\pi\) | One complete rotation |
Example 2
Convert \(150^\circ\) to radians
Multiply the degree measure by \(\dfrac{\pi}{180^\circ}\).
\[
\begin{aligned}
150^\circ
\times
\frac{\pi}{180^\circ}
&=
\frac{150\pi}{180}
\\[6pt]
&=
\frac{5\pi}{6}
\end{aligned}
\]
Answer:
\[
150^\circ=\frac{5\pi}{6}
\]
Example 3
Convert \(\dfrac{7\pi}{4}\) to degrees
Multiply the radian measure by \(\dfrac{180^\circ}{\pi}\).
\[
\begin{aligned}
\frac{7\pi}{4}
\times
\frac{180^\circ}{\pi}
&=
\frac{7(180^\circ)}{4}
\\[6pt]
&=
315^\circ
\end{aligned}
\]
Answer:
\[
\frac{7\pi}{4}=315^\circ
\]
Simplify the Units
Treat the conversion factor like a fraction. When converting to radians, the degree symbols cancel. When converting to degrees, the \(\pi\) factors cancel.
Special Angles and Exact Values
The most important unit-circle angles come from the \(30\text{-}60\text{-}90\) and \(45\text{-}45\text{-}90\) special right triangles. These triangles produce exact coordinate values without requiring a calculator.
First-Quadrant Pattern
The Three Special Angles
\(30^\circ,\ 45^\circ,\ 60^\circ\)
As the angle increases, the \(x\)-coordinate decreases while the \(y\)-coordinate increases.
\(45\text{-}45\text{-}90\)
Equal Leg Lengths
\[
1:1:\sqrt{2}
\]
After scaling the hypotenuse to \(1\), each leg has length \(\dfrac{\sqrt{2}}{2}\).
\(30\text{-}60\text{-}90\)
Three Different Side Lengths
\[
1:\sqrt{3}:2
\]
After scaling the hypotenuse to \(1\), the legs have lengths \(\dfrac{1}{2}\) and \(\dfrac{\sqrt{3}}{2}\).
First-Quadrant Unit Circle Values
| Degrees | Radians | \(\cos\theta\) | \(\sin\theta\) | Point \((\cos\theta,\sin\theta)\) |
|---|---|---|---|---|
| \(0^\circ\) | \(0\) | \(1\) | \(0\) | \((1,0)\) |
| \(30^\circ\) | \(\dfrac{\pi}{6}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | \(\left( \dfrac{\sqrt{3}}{2}, \dfrac{1}{2} \right)\) |
| \(45^\circ\) | \(\dfrac{\pi}{4}\) | \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{\sqrt{2}}{2}\) | \(\left( \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right)\) |
| \(60^\circ\) | \(\dfrac{\pi}{3}\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\left( \dfrac{1}{2}, \dfrac{\sqrt{3}}{2} \right)\) |
| \(90^\circ\) | \(\dfrac{\pi}{2}\) | \(0\) | \(1\) | \((0,1)\) |
A Useful Memory Pattern
Build the First-Quadrant Values
For the angles \(0^\circ,\ 30^\circ,\ 45^\circ,\ 60^\circ,\ 90^\circ\), the sine values follow this pattern:
\[
\sin\theta=
\frac{
\sqrt{0},
\sqrt{1},
\sqrt{2},
\sqrt{3},
\sqrt{4}
}{2}
\]
The cosine values use the same list in reverse order:
\[
\cos\theta=
\frac{
\sqrt{4},
\sqrt{3},
\sqrt{2},
\sqrt{1},
\sqrt{0}
}{2}
\]
Simplifying the square roots produces the exact values in the table.
Example 4
Find \(\sin\left(\dfrac{\pi}{3}\right)\) and \(\cos\left(\dfrac{\pi}{3}\right)\)
The angle \(\dfrac{\pi}{3}\) is equivalent to \(60^\circ\). Its unit-circle point is
\[
\left(
\frac{1}{2},
\frac{\sqrt{3}}{2}
\right).
\]
Cosine is the first coordinate and sine is the second coordinate.
Answer:
\[
\cos\left(\frac{\pi}{3}\right)
=
\frac{1}{2},
\qquad
\sin\left(\frac{\pi}{3}\right)
=
\frac{\sqrt{3}}{2}
\]
Learn the First Quadrant First
The coordinate magnitudes repeat in every quadrant. Once you know the first-quadrant values, you only need to determine which coordinates should be positive or negative.
Signs in Each Quadrant
The exact coordinate values repeat around the unit circle, but their signs change depending on the quadrant. The signs come directly from the \(x\)- and \(y\)-coordinates.
Sign Pattern
Positive and Negative Values by Quadrant
\((x,y)=(\cos\theta,\sin\theta)\)
Cosine follows the sign of \(x\), sine follows the sign of \(y\), and tangent follows the sign of \(\dfrac{y}{x}\).
Quadrant I
All Values Are Positive
\(\sin\theta>0\)
\(\cos\theta>0\)
\(\tan\theta>0\)
Both coordinates are positive, so sine, cosine, and tangent are all positive.
Quadrant II
Only Sine Is Positive
\(\sin\theta>0\)
\(\cos\theta<0\)
\(\tan\theta<0\)
The \(x\)-coordinate is negative and the \(y\)-coordinate is positive.
Quadrant III
Only Tangent Is Positive
\(\sin\theta<0\)
\(\cos\theta<0\)
\(\tan\theta>0\)
Both coordinates are negative. Their quotient is therefore positive.
Quadrant IV
Only Cosine Is Positive
\(\sin\theta<0\)
\(\cos\theta>0\)
\(\tan\theta<0\)
The \(x\)-coordinate is positive and the \(y\)-coordinate is negative.
Quadrant Sign Summary
| Quadrant | Coordinate Signs | Positive Functions |
|---|---|---|
| I | \((+,+)\) | Sine, cosine, and tangent |
| II | \((-,+)\) | Sine |
| III | \((-,-)\) | Tangent |
| IV | \((+,-)\) | Cosine |
Example 5
Determine the signs at \(150^\circ\)
An angle of \(150^\circ\) lies in Quadrant II. In Quadrant II, the \(x\)-coordinate is negative and the \(y\)-coordinate is positive.
\[
\cos(150^\circ)<0
\]
\[
\sin(150^\circ)>0
\]
\[
\tan(150^\circ)<0
\]
Answer:
Sine is positive, while cosine and
tangent are negative.
Optional Memory Aid
All Students Take Calculus
All
Quadrant I
Sine
Quadrant II
Tangent
Quadrant III
Cosine
Quadrant IV
This phrase identifies which trigonometric functions are positive in each quadrant. The coordinate signs explain why the pattern works.
Using Reference Angles
A reference angle is the positive acute angle formed between the terminal side of an angle and the \(x\)-axis. Reference angles let us reuse the special-angle values from Quadrant I.
Reference Angle Rule
First find the related acute angle. Then use the original angle’s quadrant to determine whether each trigonometric value is positive or negative.
Visual Model
An Angle and Its Reference Angle
Example: \(150^\circ\)
The terminal side of \(150^\circ\) is \(30^\circ\) away from the negative \(x\)-axis, so its reference angle is \(30^\circ\).
Quadrant I
Use the Original Angle
\[
\theta_{\text{ref}}=\theta
\]
Angles in Quadrant I are already positive acute angles.
Quadrant II
Subtract From \(180^\circ\)
\[
\theta_{\text{ref}}
=
180^\circ-\theta
\]
In radians, use \(\pi-\theta\).
Quadrant III
Subtract \(180^\circ\)
\[
\theta_{\text{ref}}
=
\theta-180^\circ
\]
In radians, use \(\theta-\pi\).
Quadrant IV
Subtract From \(360^\circ\)
\[
\theta_{\text{ref}}
=
360^\circ-\theta
\]
In radians, use \(2\pi-\theta\).
Example 6
Find the exact value of \(\sin(150^\circ)\)
The angle \(150^\circ\) lies in Quadrant II. Find its reference angle.
\[
\begin{aligned}
\theta_{\text{ref}}
&=
180^\circ-150^\circ
\\[4pt]
&=
30^\circ
\end{aligned}
\]
The sine value for \(30^\circ\) is \(\dfrac{1}{2}\). Sine is positive in Quadrant II.
Answer:
\[
\sin(150^\circ)
=
\sin(30^\circ)
=
\frac{1}{2}
\]
Example 7
Find the exact value of \(\cos\left(\dfrac{5\pi}{4}\right)\)
The angle \(\dfrac{5\pi}{4}\) lies in Quadrant III. Subtract \(\pi\) to find its reference angle.
\[
\begin{aligned}
\theta_{\text{ref}}
&=
\frac{5\pi}{4}-\pi
\\[6pt]
&=
\frac{5\pi}{4}
-
\frac{4\pi}{4}
\\[6pt]
&=
\frac{\pi}{4}
\end{aligned}
\]
The cosine value for \(\dfrac{\pi}{4}\) has magnitude \(\dfrac{\sqrt{2}}{2}\). Cosine is negative in Quadrant III.
Answer:
\[
\cos\left(\frac{5\pi}{4}\right)
=
-\frac{\sqrt{2}}{2}
\]
Reference-Angle Process
- Identify the quadrant containing the original angle.
- Find the positive acute reference angle.
- Use the matching first-quadrant exact value.
- Apply the correct sign based on the original quadrant.
Finding Tangent Values on the Unit Circle
Tangent is not one of the coordinates of a unit-circle point. Instead, tangent is the ratio of the sine value to the cosine value.
Tangent Relationship
\[
\tan\theta
=
\frac{\sin\theta}{\cos\theta}
=
\frac{y}{x}
\]
Once you know the coordinates \((x,y)\), divide the second coordinate by the first coordinate to find tangent.
Positive Tangent
Coordinates Have the Same Sign
Tangent is positive in Quadrants I and III because \(x\) and \(y\) have the same sign.
\[
\frac{+}{+}=+
\qquad
\frac{-}{-}=+
\]
Negative Tangent
Coordinates Have Opposite Signs
Tangent is negative in Quadrants II and IV because \(x\) and \(y\) have opposite signs.
\[
\frac{+}{-}=-
\qquad
\frac{-}{+}=-
\]
Common First-Quadrant Tangent Values
| Angle | Sine | Cosine | Tangent |
|---|---|---|---|
| \(0^\circ\) | \(0\) | \(1\) | \(0\) |
| \(30^\circ\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{\sqrt{3}}{3}\) |
| \(45^\circ\) | \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{\sqrt{2}}{2}\) | \(1\) |
| \(60^\circ\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | \(\sqrt{3}\) |
| \(90^\circ\) | \(1\) | \(0\) | Undefined |
Example 8
Find \(\tan\left(\dfrac{2\pi}{3}\right)\)
The angle \(\dfrac{2\pi}{3}\) is \(120^\circ\), which lies in Quadrant II. Its unit-circle point is
\[
\left(
-\frac{1}{2},
\frac{\sqrt{3}}{2}
\right).
\]
Divide the \(y\)-coordinate by the \(x\)-coordinate.
\[
\begin{aligned}
\tan\left(\frac{2\pi}{3}\right)
&=
\frac{
\frac{\sqrt{3}}{2}
}{
-\frac{1}{2}
}
\\[6pt]
&=
-\sqrt{3}
\end{aligned}
\]
Answer:
\[
\tan\left(\frac{2\pi}{3}\right)
=
-\sqrt{3}
\]
Example 9
Why is \(\tan\left(\dfrac{\pi}{2}\right)\) undefined?
At \(\dfrac{\pi}{2}\), the unit-circle point is \((0,1)\).
\[
\tan\left(\frac{\pi}{2}\right)
=
\frac{y}{x}
=
\frac{1}{0}
\]
Division by zero is undefined.
Answer:
\[
\tan\left(\frac{\pi}{2}\right)
\text{ is undefined.}
\]
Tangent Is Undefined When Cosine Is Zero
Because \(\tan\theta= \dfrac{\sin\theta}{\cos\theta}\), tangent is undefined wherever \(\cos\theta=0\). On the unit circle, this occurs at \(\dfrac{\pi}{2}\) and \(\dfrac{3\pi}{2}\).
Key Reminders
Avoid Common Unit Circle Mistakes
Most unit-circle errors come from reversing coordinates, mixing degrees with radians, or forgetting how signs change between quadrants.
1
Reversing Sine and Cosine
The coordinates are always written with cosine first and sine second.
\[
(x,y)
=
(\cos\theta,\sin\theta)
\]
Think: cosine is horizontal and sine is vertical.
2
Mixing Degrees and Radians
An angle such as \(60^\circ\) and an angle such as \(\dfrac{\pi}{3}\) represent the same rotation, but they use different units.
\[
60^\circ=\frac{\pi}{3}
\]
Check the unit before choosing a conversion or identifying an angle.
3
Ignoring the Quadrant
A reference angle gives the magnitude of a value. The quadrant determines its sign.
\[
\cos(120^\circ)
=
-\cos(60^\circ)
=
-\frac{1}{2}
\]
Identify the quadrant before writing the final answer.
4
Using the Angle Instead of the Reference Angle
The special-triangle values come from acute reference angles such as \(30^\circ\), \(45^\circ\), and \(60^\circ\).
\[
210^\circ-180^\circ
=
30^\circ
\]
Reduce the original angle to its related first-quadrant angle.
5
Treating Tangent as a Coordinate
A unit-circle point directly gives cosine and sine. Tangent must be calculated as a ratio.
\[
\tan\theta
=
\frac{y}{x}
\]
Divide the second coordinate by the first coordinate.
6
Dividing by Zero
Tangent is undefined when the \(x\)-coordinate, or cosine, is zero.
\[
\tan\left(\frac{\pi}{2}\right)
=
\frac{1}{0}
\text{ is undefined}
\]
Never report a numerical tangent value when cosine equals zero.
Final Unit Circle Check
- Did you identify whether the angle is in degrees or radians?
- Did you place the angle in the correct quadrant?
- Did you use cosine for the \(x\)-coordinate and sine for the \(y\)-coordinate?
- Did you apply the correct positive or negative signs?
- If finding tangent, did you calculate \(y/x\)?
Complete Reference
The Unit Circle at a Glance
The complete unit circle combines degree measures, radian measures, and coordinates. Each coordinate is written in the order \((\cos\theta,\sin\theta)\).
Complete Unit Circle
Special Angles and Exact Coordinates
\((x,y)=(\cos\theta,\sin\theta)\)
The coordinate magnitudes repeat in each quadrant. Only the signs change.
Special Angles Around the Circle
Download Reference PDF| Degrees | Radians | \(\cos\theta\) | \(\sin\theta\) | \(\tan\theta\) |
|---|---|---|---|---|
| \(0^\circ\) | \(0\) | \(1\) | \(0\) | \(0\) |
| \(30^\circ\) | \(\dfrac{\pi}{6}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{3}\) |
| \(45^\circ\) | \(\dfrac{\pi}{4}\) | \(\dfrac{\sqrt{2}}{2}\) | \(\dfrac{\sqrt{2}}{2}\) | \(1\) |
| \(60^\circ\) | \(\dfrac{\pi}{3}\) | \(\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | \(\sqrt{3}\) |
| \(90^\circ\) | \(\dfrac{\pi}{2}\) | \(0\) | \(1\) | Undefined |
| \(120^\circ\) | \(\dfrac{2\pi}{3}\) | \(-\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{2}\) | \(-\sqrt{3}\) |
| \(135^\circ\) | \(\dfrac{3\pi}{4}\) | \(-\dfrac{\sqrt{2}}{2}\) | \(\dfrac{\sqrt{2}}{2}\) | \(-1\) |
| \(150^\circ\) | \(\dfrac{5\pi}{6}\) | \(-\dfrac{\sqrt{3}}{2}\) | \(\dfrac{1}{2}\) | \(-\dfrac{\sqrt{3}}{3}\) |
| \(180^\circ\) | \(\pi\) | \(-1\) | \(0\) | \(0\) |
| \(210^\circ\) | \(\dfrac{7\pi}{6}\) | \(-\dfrac{\sqrt{3}}{2}\) | \(-\dfrac{1}{2}\) | \(\dfrac{\sqrt{3}}{3}\) |
| \(225^\circ\) | \(\dfrac{5\pi}{4}\) | \(-\dfrac{\sqrt{2}}{2}\) | \(-\dfrac{\sqrt{2}}{2}\) | \(1\) |
| \(240^\circ\) | \(\dfrac{4\pi}{3}\) | \(-\dfrac{1}{2}\) | \(-\dfrac{\sqrt{3}}{2}\) | \(\sqrt{3}\) |
| \(270^\circ\) | \(\dfrac{3\pi}{2}\) | \(0\) | \(-1\) | Undefined |
| \(300^\circ\) | \(\dfrac{5\pi}{3}\) | \(\dfrac{1}{2}\) | \(-\dfrac{\sqrt{3}}{2}\) | \(-\sqrt{3}\) |
| \(315^\circ\) | \(\dfrac{7\pi}{4}\) | \(\dfrac{\sqrt{2}}{2}\) | \(-\dfrac{\sqrt{2}}{2}\) | \(-1\) |
| \(330^\circ\) | \(\dfrac{11\pi}{6}\) | \(\dfrac{\sqrt{3}}{2}\) | \(-\dfrac{1}{2}\) | \(-\dfrac{\sqrt{3}}{3}\) |
| \(360^\circ\) | \(2\pi\) | \(1\) | \(0\) | \(0\) |
Do Not Memorize 17 Unrelated Rows
Learn the five first-quadrant points, reflect their coordinate magnitudes into the other quadrants, and apply the appropriate signs. The angle patterns then repeat around the circle.
Guide Summary
What to Remember About the Unit Circle
The unit circle has radius \(1\) and is centered at \((0,0)\).
A point at angle \(\theta\) is \((\cos\theta,\sin\theta)\).
Degrees and radians describe the same rotations using different units.
First-quadrant exact values come from special right triangles.
Reference angles determine value magnitudes, while quadrants determine signs.
Tangent is \(\dfrac{\sin\theta}{\cos\theta}\) and is undefined when cosine is zero.
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