From Unit-Circle Coordinates to Tangent
For an angle \(\theta\), the unit-circle point has coordinates \((\cos\theta,\sin\theta)\). Sine is its vertical coordinate and cosine is its horizontal coordinate. Tangent combines these two coordinates in a quotient: the vertical coordinate divided by the horizontal coordinate.
This definition gives a direct way to calculate tangent, but it also imposes a condition. Division by zero is undefined, so tangent has no value at angles whose unit-circle point lies on the vertical axis. The signs of the two coordinates also determine whether tangent is positive or negative.
The condition \(\cos\theta\ne 0\) is part of the definition, not an optional check. Before calculating a tangent value, first make sure that the denominator is nonzero.
Why the Quotient Is a Slope
The quotient also has a simple geometric interpretation. Let \(P=(x,y)\) be the unit-circle point for an angle \(\theta\), measured from the positive horizontal axis. If \(x\ne 0\), the ray from the origin through \(P\) has slope equal to its vertical change divided by its horizontal change.
By the unit-circle coordinate definitions, \(x=\cos\theta\) and \(y=\sin\theta\). Substituting these coordinates gives
This explains both the quotient and its restriction. When \(x=0\), the ray is vertical and its slope is undefined; at the same time, \(\cos\theta=x=0\), so the quotient defining tangent has a zero denominator. Here “slope” means the slope of the ray from the origin to the unit-circle point, not the slope of a line tangent to the circle.
Where Tangent Is Undefined
Tangent is undefined precisely when the cosine coordinate is zero. On the unit circle, that happens at points on the vertical axis, \((0,1)\) and \((0,-1)\). In radians, these points correspond to angles of the form
For example, \(\pi/2\) and \(3\pi/2\) correspond to the top and bottom of the unit circle. At either point, sine is nonzero but cosine is zero, so the quotient cannot be evaluated. Saying that tangent is “infinite” at such an angle is not correct: division by zero is undefined, and there is no real tangent value there.
At angles not on the vertical axis, the horizontal coordinate is nonzero, so the quotient is defined. This domain condition applies regardless of whether the angle is positive or negative, or how many turns it makes around the circle.
Signs from the Unit-Circle Coordinates
When tangent is defined, its sign follows from the signs of sine and cosine. A quotient of two positive numbers is positive, and a quotient of two negative numbers is also positive. A quotient with one positive and one negative number is negative.
| Location of the point | Sign of sine | Sign of cosine | Sign of tangent |
|---|---|---|---|
| First quadrant | Positive | Positive | Positive |
| Second quadrant | Positive | Negative | Negative |
| Third quadrant | Negative | Negative | Positive |
| Fourth quadrant | Negative | Positive | Negative |
This sign check is useful before simplifying an exact value. For instance, an angle in the second quadrant must have negative tangent: its sine coordinate is positive and its cosine coordinate is negative. On the horizontal axis, the sine coordinate is zero and the cosine coordinate is nonzero, so tangent is zero. On the vertical axis, tangent is undefined instead.
Worked Examples
Worked Example: Tangent in the First Quadrant
Find \(\tan(\pi/3)\), using the unit-circle coordinates for \(\pi/3\).
At \(\theta=\pi/3\), the point is \((\cos\theta,\sin\theta)=(1/2,\sqrt{3}/2)\). In particular, \(\cos\theta=1/2\ne0\), so tangent is defined. Apply the quotient definition:
The coordinates are both positive, so the positive result has the expected sign. Check the quotient by multiplying the answer by the denominator: \(\sqrt{3}\cdot(1/2)=\sqrt{3}/2\), which recovers the numerator. Therefore \(\tan(\pi/3)=\sqrt{3}\).
Worked Example: A Negative Tangent in the Second Quadrant
Find \(\tan(5\pi/6)\) from the unit-circle coordinates.
The angle \(5\pi/6\) lies in the second quadrant. Its reference angle is \(\pi/6\), so its coordinates are \((-\sqrt{3}/2,1/2)\): the horizontal coordinate is negative and the vertical coordinate is positive. Thus \(\cos(5\pi/6)=-\sqrt{3}/2\ne0\), and the quotient is defined.
The last equality comes from multiplying numerator and denominator by \(\sqrt{3}\). The signs differ, so tangent must be negative, as found. Check by multiplying the answer by the cosine coordinate: \((-\sqrt{3}/3)(-\sqrt{3}/2)=3/6=1/2\), which recovers the sine coordinate. Hence \(\tan(5\pi/6)=-\sqrt{3}/3\).
Worked Example: Tangent Undefined on the Vertical Axis
Determine whether \(\tan(3\pi/2)\) is defined.
At \(3\pi/2\), the unit-circle point is \((0,-1)\). Therefore \(\cos(3\pi/2)=0\) and \(\sin(3\pi/2)=-1\). Substitution into the quotient definition would require division by zero:
Since division by zero is undefined, \(\tan(3\pi/2)\) is undefined. The vertical-ray interpretation gives the same conclusion: the ray through \((0,-1)\) is vertical and has no defined slope. This is not a value of zero or an infinite real number.
A Reliable Calculation Process
When an angle is given in a form for which you know its unit-circle coordinates, use the following sequence. The order matters: checking the denominator first prevents an undefined quotient from being treated as an ordinary calculation.
Write the coordinates as \((\cos\theta,\sin\theta)\), using the quadrant and any relevant reference angle.
If \(\cos\theta=0\), stop: tangent is undefined. Otherwise the quotient can be calculated.
Evaluate \(\sin\theta/\cos\theta\), preserving the signs of both coordinates.
Confirm that its sign agrees with the quadrant and that multiplying it by \(\cos\theta\) gives \(\sin\theta\).
Common Mistake: Reversing the Quotient
A frequent error is to divide the horizontal coordinate by the vertical coordinate. Tangent is vertical divided by horizontal, not horizontal divided by vertical. The coordinate order in \((\cos\theta,\sin\theta)\) is horizontal first and vertical second, but the slope formula is vertical change over horizontal change.
Worked Example: Correcting a Reversed Quotient
Consider \(\theta=\pi/4\), whose unit-circle coordinates are \((\sqrt{2}/2,\sqrt{2}/2)\). A reversed quotient would give \[ \frac{\cos(\pi/4)}{\sin(\pi/4)} = \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1. \] Here that incorrect method happens to produce the correct numerical result because the two coordinates are equal. That coincidence can hide the mistake.
Use a different angle to expose it. For \(\theta=\pi/3\), the coordinates are \((1/2,\sqrt{3}/2)\). The reversed calculation gives \[ \frac{\cos(\pi/3)}{\sin(\pi/3)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}, \] but the definition requires sine divided by cosine:
Check the correct answer: \(\sqrt{3}\cdot(1/2)=\sqrt{3}/2\), the sine coordinate. The reversed answer instead gives \((1/\sqrt{3})(1/2)=1/(2\sqrt{3})\), which is not \(\sqrt{3}/2\). The correction is to keep the slope order: vertical coordinate over horizontal coordinate.
Why the Coordinate View Is Useful
The coordinate definition makes tangent calculations systematic. Sine and cosine provide the coordinates; tangent is their quotient when the horizontal coordinate is nonzero. This lets you determine both existence and sign before doing any simplification.
The slope interpretation provides an additional check. A point in the first or third quadrant lies along a ray with positive slope, while a point in the second or fourth quadrant lies along a ray with negative slope. A point on the vertical axis has no finite slope, matching the zero-denominator restriction. These conclusions follow directly from the coordinate signs and do not require memorizing separate tangent values for each quadrant.
When using exact unit-circle values, keep the coordinates exact through the division. In particular, retain the negative sign of a coordinate in the second or fourth quadrant. The final multiplication check is simple: a proposed tangent value \(t\) must satisfy \(t\cos\theta=\sin\theta\), with \(\cos\theta\ne0\).
Check Your Understanding
Use unit-circle coordinates and the quotient definition to answer these questions.
- What condition on \(\cos\theta\) is required for \(\tan\theta\) to be defined?
- At which points on the unit circle is tangent undefined, and why?
- If \(\sin\theta<0\) and \(\cos\theta<0\), what is the sign of \(\tan\theta\)?
- Find \(\tan(7\pi/4)\) using its unit-circle coordinates, and check the answer by multiplication.
- In the coordinate interpretation, what quantity does tangent give when it is defined?
