Tangent and Reciprocal Functions

Trigonometry involves six fundamental functions, but most introductory courses begin with only sine and cosine. The remaining four — tangent, cotangent, secant, and cosecant — are equally important and appear constantly in calculus, physics, and engineering. Three of these four are defined as ratios or reciprocals of sine and cosine, which is why they are collectively called the reciprocal and quotient trigonometric functions. The tangent function relates the opposite and adjacent sides of a right triangle and has a distinctive graph with vertical asymptotes. Secant is the reciprocal of cosine, cosecant is the reciprocal of sine, and cotangent is the reciprocal of tangent (equivalently, cosine divided by sine). Each function has its own domain restrictions, period, and set of identities that make it useful in different mathematical contexts. Mastering these four functions means understanding not just their definitions but also their graphs, their undefined values (where denominators equal zero), and how they relate to the unit circle. Together with sine and cosine, they complete the full picture of circular trigonometry and form the foundation for more advanced topics like inverse trigonometric functions, trigonometric identities, and calculus derivatives.

Key takeaways
  • Tangent as a Ratio
  • Cotangent
  • Secant
  • Cosecant
  • Key Pythagorean Identities Involving These Functions
  • Evaluating at Common Angles

Tangent as a Ratio

For an angle θ in a right triangle, tangent is defined as the ratio of the opposite side to the adjacent side: tan θ = opposite/adjacent. On the unit circle (radius = 1), with a point (x, y) on the circle at angle θ, this becomes tan θ = y/x. This means tan θ is undefined whenever x = cos θ = 0, which occurs at θ = π/2 + nπ for any integer n.

Cotangent

Cotangent is the reciprocal of tangent: cot θ = 1/tan θ = cos θ / sin θ = adjacent/opposite. It is undefined wherever sin θ = 0, that is, at θ = nπ for integer n. Note that cot θ ≠ tan(−θ); cotangent is its own distinct function with a period of π, just like tangent.

Secant

Secant is the reciprocal of cosine: sec θ = 1/cos θ. It is undefined wherever cos θ = 0, i.e., at θ = π/2 + nπ. The range of secant is (−∞, −1] ∪ [1, ∞) because |cos θ| ≤ 1 means |sec θ| ≥ 1. Secant has a period of 2π, matching cosine.

Cosecant

Cosecant is the reciprocal of sine: csc θ = 1/sin θ. It is undefined wherever sin θ = 0, i.e., at θ = nπ. Like secant, its range is (−∞, −1] ∪ [1, ∞). Cosecant has a period of 2π, matching sine.

Key Pythagorean Identities Involving These Functions

Starting from sin²θ + cos²θ = 1 and dividing through by cos²θ gives tan²θ + 1 = sec²θ. Dividing by sin²θ gives 1 + cot²θ = csc²θ. These two identities are essential tools for simplifying expressions and solving equations. For example, if sec θ = 3, then tan²θ = sec²θ − 1 = 9 − 1 = 8, so tan θ = ±2√2.

Evaluating at Common Angles

At θ = π/6 (30°): tan(π/6) = sin(π/6)/cos(π/6) = (1/2)/(√3/2) = 1/√3 = √3/3; sec(π/6) = 2/√3 = 2√3/3; csc(π/6) = 2; cot(π/6) = √3. At θ = π/4 (45°): tan(π/4) = 1; sec(π/4) = √2; csc(π/4) = √2; cot(π/4) = 1. At θ = π/3 (60°): tan(π/3) = √3; sec(π/3) = 2; csc(π/3) = 2√3/3; cot(π/3) = 1/√3 = √3/3.

The Key Equation

tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ,sec⁡θ=1cos⁡θ,csc⁡θ=1sin⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}
  • \tan\theta — Tangent (dimensionless): Ratio of sine to cosine; undefined at θ = π/2 + nπ; period π
  • \cot\theta — Cotangent (dimensionless): Ratio of cosine to sine; undefined at θ = nπ; period π
  • \sec\theta — Secant (dimensionless): Reciprocal of cosine; range (−∞,−1]∪[1,∞); period 2π
  • \csc\theta — Cosecant (dimensionless): Reciprocal of sine; range (−∞,−1]∪[1,∞); period 2π

Worked Example

Problem. Given that tan θ = −4/3 and θ is in the second quadrant (π/2 < θ < π), find the exact values of all six trigonometric functions.

sin⁡θ=45, cos⁡θ=−35, tan⁡θ=−43, csc⁡θ=54, sec⁡θ=−53, cot⁡θ=−34\sin\theta=\tfrac{4}{5},\ \cos\theta=-\tfrac{3}{5},\ \tan\theta=-\tfrac{4}{3},\ \csc\theta=\tfrac{5}{4},\ \sec\theta=-\tfrac{5}{3},\ \cot\theta=-\tfrac{3}{4}

Answer. sin θ = 4/5, cos θ = −3/5, tan θ = −4/3, csc θ = 5/4, sec θ = −5/3, cot θ = −3/4

Figure
Diagram
Graphs of tan θ and sec θ on [−π, π]θy−π/2π/2−ππ02−21−1y = tan θy = sec θ--- asymptotes
Origins · 1 minAbu al-Wafa' al-Buzjani and how this was discovered
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Abu al-Wafa' al-Buzjani
940–998 · Persian

Abu al-Wafa' was a 10th-century Persian mathematician and astronomer who made pioneering contributions to trigonometry. He is credited with introducing the tangent function into trigonometry and constructing more accurate trigonometric tables than had existed previously. He also developed the secant and cosecant functions and applied them systematically to astronomical calculations, effectively establishing all six trigonometric functions as a complete system. Abu al-Wafa' computed a table of sines at intervals of 15 arc minutes with an accuracy to 8 sexagesimal places — a remarkable computational feat for the 10th century, achieved without modern algebraic notation.

  • Sine and Cosine Functions — Tangent, secant, cosecant, and cotangent are all defined directly in terms of sine and cosine, making a solid understanding of these two primary functions essential.
  • The Unit Circle — The unit circle provides the geometric foundation for evaluating all six trigonometric functions at any angle, including identifying where the reciprocal functions are undefined.
  • Trigonometric Identities — The Pythagorean identities tan²θ + 1 = sec²θ and 1 + cot²θ = csc²θ are fundamental tools that follow directly from the definitions of these four functions.
  • Inverse Trigonometric Functions — Each of the six trigonometric functions, including tangent and secant, has an inverse function (arctan, arcsec, etc.) used extensively in calculus and problem solving.