Difficulty: Intermediate | Prerequisites: Unit circle values, sine and cosine graphs, concept of asymptotes
Tangent and cotangent are the two trigonometric functions built from ratios of sine and cosine. Unlike sine and cosine, they have vertical asymptotes and no amplitude, which makes their graphs look and behave quite differently. If you are comfortable reading sine and cosine graphs and know your unit circle values, this topic will click quickly. If not, revise those first. This material appears in nearly every precalculus and trigonometry course and forms the foundation for understanding all trig-function transformations.
The tangent function equals sin(x)/cos(x), so it blows up wherever cosine is zero. The cotangent function equals cos(x)/sin(x), so it blows up wherever sine is zero. Both have period π, range of all real numbers, and their asymptotes and x-intercepts are swapped versions of each other.
Tangent function, tan(x)
The ratio sin(x)/cos(x). It gives the slope of the terminal side of angle x on the unit circle. In simple terms, tangent tells you how steeply a radius line tilts at any given angle.
Cotangent function, cot(x)
The ratio cos(x)/sin(x), which is the reciprocal of tangent. Think of it as tangent flipped: wherever tangent is 1, cotangent is 1, but their zeros and undefined points trade places.
Vertical asymptote
A vertical line that the graph approaches but never touches or crosses. For tan(x), these occur where cos(x) = 0. For cot(x), they occur where sin(x) = 0. In simple terms, these are the "walls" the curve races towards at the edges of each period.
Period
The horizontal length of one complete cycle of the function before it repeats. Both parent tan(x) and cot(x) have a period of π, which is half the period of sine and cosine.
Undefined value
A point where the function has no output because division by zero would occur. Tangent is undefined at x = π/2 + nπ; cotangent is undefined at x = nπ (where n is any integer).
tan(x) = sin(x) / cos(x)
Wherever cos(x) = 0, tangent is undefined, producing a vertical asymptote
The function passes through the origin: tan(0) = 0
Key reference values in one period (0 to π/2):
tan(0) = 0
tan(π/6) = √3/3 ≈ 0.577
tan(π/4) = 1
tan(π/3) = √3 ≈ 1.732
tan(π/2) = undefined
The function is odd, so tan(−x) = −tan(x). Negative angle values mirror across the origin.
Domain: all real numbers except x = π/2 + nπ, where n is any integer
Range: all real numbers (the curve stretches to +∞ and −∞ within every period)
Period: π
Vertical asymptotes: x = π/2 + nπ
x-intercepts: x = nπ (the curve crosses the x-axis exactly at integer multiples of π)
Shape: within each period the curve rises from −∞ to +∞, passing through the midpoint at y = 0. It looks like an elongated S turned on its side, repeating every π units.
cot(x) = cos(x) / sin(x), which equals 1/tan(x)
Wherever sin(x) = 0, cotangent is undefined, producing a vertical asymptote
Key reference values:
cot(0) = undefined
cot(π/6) = √3 ≈ 1.732
cot(π/4) = 1
cot(π/3) = √3/3 ≈ 0.577
cot(π/2) = 0
Domain: all real numbers except x = nπ, where n is any integer
Range: all real numbers
Period: π
Vertical asymptotes: x = nπ
x-intercepts: x = π/2 + nπ
Shape: within each period the curve falls from +∞ to −∞. It is a decreasing version of the tangent shape.
This is the single most useful pattern to remember: the asymptotes of tangent sit exactly where the x-intercepts of cotangent are, and vice versa. If you know one function's graph, you can locate every key feature of the other by swapping zeros and asymptotes.
Property | tan(x) | cot(x) |
|---|---|---|
Definition | sin(x)/cos(x) | cos(x)/sin(x) |
Period | π | π |
Asymptotes | x = π/2 + nπ | x = nπ |
x-intercepts | x = nπ | x = π/2 + nπ |
Range | All real numbers | All real numbers |
Passes through | (0, 0) and (π, 0) | (π/2, 0) and (3π/2, 0) |
Tangent shows up whenever you need the slope or gradient of a line at an angle, which is why surveyors and engineers use it constantly to calculate heights and distances from angle measurements. Cotangent appears in the same settings when the complementary angle is more convenient to work with.
Students often think tangent and cotangent have amplitude like sine and cosine. They do not. Because their range is all real numbers, there is no maximum or minimum value and therefore no amplitude to speak of.
Students sometimes memorise asymptote locations by rote without understanding why they exist. The asymptotes come directly from division by zero in the sin/cos ratio. If you forget the locations, just ask "where does the denominator equal zero?" and you have your answer.
Mixing up which function has asymptotes at nπ versus π/2 + nπ is extremely common. Remember: tangent's denominator is cosine, and cosine equals zero at π/2 + nπ. Cotangent's denominator is sine, and sine equals zero at nπ.
Students sometimes assume the period is 2π because sine and cosine have that period. Tangent and cotangent repeat every π, not 2π.
⚠️ You will almost certainly be asked to identify asymptotes, domain, range, and period from a graph or equation. Have the parent-function properties memorised cold.
⚠️ Expect a question that asks you to explain the relationship between tangent and cotangent (the asymptote/intercept swap). This is a favourite short-answer item.
⚠️ Knowing the five key reference points (0, π/6, π/4, π/3, π/2) for tangent lets you sketch the curve by hand without a calculator. Exams often require a hand-sketch.
True or false: The range of y = tan(x) is [−1, 1].
Fill in the blank: The period of the parent cotangent function is ___.
True or false: cot(x) has a vertical asymptote at x = π/2.
Fill in the blank: tan(x) is undefined wherever ___(x) = 0.
True or false: The x-intercepts of tan(x) occur at the same x-values as the asymptotes of cot(x).
Answers: 1. False (range is all real numbers). 2. π. 3. False (its asymptotes are at x = nπ, so x = π/2 is an intercept). 4. cos. 5. True.
Q: State the domain, range, period, and asymptote locations for y = tan(x).
A: Domain is all real numbers except x = π/2 + nπ (n any integer). Range is all real numbers. Period is π. Vertical asymptotes at x = π/2 + nπ.
Q: Where are the x-intercepts of y = cot(x), and why?
A: At x = π/2 + nπ, because cot(x) = cos(x)/sin(x) and cos(x) = 0 at those values, making the numerator zero (while the denominator is non-zero).
Q: Explain why tangent has vertical asymptotes at x = π/2 + nπ.
A: Because tan(x) = sin(x)/cos(x), and cos(x) = 0 at x = π/2 + nπ. Division by zero produces an undefined value, and the function approaches ±∞ near those points, creating vertical asymptotes.
Q: How do the graphs of tangent and cotangent relate to each other in terms of increasing/decreasing behaviour?
A: Within each period, tangent is always increasing (rises from −∞ to +∞). Cotangent is always decreasing (falls from +∞ to −∞). They are, in a sense, mirror images of each other in terms of direction.
This material connects directly to graphing sine and cosine, since tangent and cotangent are built from those functions. Understanding asymptotes here prepares you for secant and cosecant graphs, which also have vertical asymptotes driven by the same zero-denominator logic. Transformations of tangent and cotangent (covered in Part 2) use the same a, b, h, k framework you already know from transforming sine and cosine.
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