Definitions of Trigonometric Ratios

Trigonometric ratios are six fundamental relationships between the angles and side lengths of a right triangle. These ratios — sine, cosine, tangent, cosecant, secant, and cotangent — form the backbone of trigonometry and appear throughout mathematics, physics, engineering, and architecture. Understanding them unlocks the ability to calculate unknown side lengths and angles in any triangle, not just right triangles. There are two complementary ways to define trigonometric ratios: through right triangles and through the unit circle. The right-triangle definition is intuitive and practical for solving geometric problems, while the unit circle definition extends the ratios to all real numbers, including angles greater than 90° or even negative angles. Both definitions agree perfectly for acute angles, and together they give a complete picture of trigonometric functions. The word 'trigonometry' comes from the Greek words for triangle (trigonon) and measurement (metron). Ancient Greek and Indian mathematicians developed these ideas over two millennia ago to solve problems in astronomy and land surveying. Today, trigonometric ratios are used to model waves, oscillations, and cycles in virtually every branch of science and engineering.

Key takeaways
  • Setting Up a Right Triangle
  • The Primary Ratios: Sine, Cosine, Tangent
  • The Reciprocal Ratios: Cosecant, Secant, Cotangent
  • The Unit Circle Definition
  • Key Values to Memorize

Setting Up a Right Triangle

Consider a right triangle with one acute angle θ. Label the three sides relative to θ: the side directly opposite θ is called the opposite side (opp), the side next to θ that is not the hypotenuse is called the adjacent side (adj), and the longest side (across from the right angle) is the hypotenuse (hyp). The Pythagorean theorem guarantees that opp² + adj² = hyp².

The Primary Ratios: Sine, Cosine, Tangent

The three primary ratios are defined as: sin θ = opp/hyp, cos θ = adj/hyp, and tan θ = opp/adj. A classic mnemonic is SOH-CAH-TOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent). Note that tan θ = sin θ / cos θ, which follows directly from these definitions.

The Reciprocal Ratios: Cosecant, Secant, Cotangent

The three reciprocal ratios are simply the reciprocals of the primary ones: csc θ = hyp/opp = 1/sin θ, sec θ = hyp/adj = 1/cos θ, and cot θ = adj/opp = 1/tan θ. These are useful in many calculus and physics formulas. Remember that csc is the reciprocal of sin (not cos), which is a common source of confusion.

The Unit Circle Definition

Place an angle θ in standard position (vertex at the origin, initial side along the positive x-axis) and draw a ray. Where this ray intersects the unit circle (radius = 1, centered at origin), the intersection point is (cos θ, sin θ). This definition extends trig ratios to any angle: cos θ is the x-coordinate and sin θ is the y-coordinate of the point on the unit circle. Since the radius is 1, this is consistent with the right-triangle definition for 0° < θ < 90°, and it works for all other angles too.

Key Values to Memorize

For the standard angles, the exact values are: sin 0° = 0, cos 0° = 1; sin 30° = 1/2, cos 30° = √3/2; sin 45° = √2/2, cos 45° = √2/2; sin 60° = √3/2, cos 60° = 1/2; sin 90° = 1, cos 90° = 0. A useful memory pattern: sin of 0°, 30°, 45°, 60°, 90° equals √0/2, √1/2, √2/2, √3/2, √4/2 respectively (simplify each).

The Key Equation

sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}
  • \theta — theta (degrees or radians): The acute angle in the right triangle for which we are computing the ratio.
  • \text{opp} — opposite (length units): The side of the right triangle directly across from angle θ.
  • \text{adj} — adjacent (length units): The side of the right triangle next to angle θ that is not the hypotenuse.
  • \text{hyp} — hypotenuse (length units): The longest side of the right triangle, always opposite the 90° angle.

Worked Example

Problem. A right triangle has legs of length 5 and 12. Find all six trigonometric ratios for the angle θ opposite the side of length 5.

sin⁡θ=513,  cos⁡θ=1213,  tan⁡θ=512,  csc⁡θ=135,  sec⁡θ=1312,  cot⁡θ=125\sin\theta = \tfrac{5}{13},\; \cos\theta = \tfrac{12}{13},\; \tan\theta = \tfrac{5}{12},\; \csc\theta = \tfrac{13}{5},\; \sec\theta = \tfrac{13}{12},\; \cot\theta = \tfrac{12}{5}

Answer. sin θ = 5/13, cos θ = 12/13, tan θ = 5/12, csc θ = 13/5, sec θ = 13/12, cot θ = 12/5

Figure
Diagram
Right Triangle Trigonometric Ratiosadjacent = 12opposite = 5hypotenuse = 13θsin θ = 5/13 cos θ = 12/13 tan θ = 5/12
Origins · 1 minAryabhata and how this was discovered
A
Aryabhata
476–550 · Indian

Aryabhata was one of the first mathematicians to define and systematically use sine as a ratio (which he called 'jya'), laying the foundation for modern trigonometry. His work Aryabhatiya (499 CE) contained a table of sine values computed at intervals of 3.75°, accurate to an impressive degree for its time. His concept of the half-chord (equivalent to our sine) was a major advance over the full-chord methods used by earlier Greek astronomers. Aryabhata correctly calculated the approximate value of π as 3.1416 and knew that Earth rotates on its axis — over a thousand years before Copernicus proposed the heliocentric model in Europe.

  • Pythagorean Identities — The identity sin²θ + cos²θ = 1 follows directly from the definitions of sine and cosine and the Pythagorean theorem — it is the algebraic consequence of placing the ratios on the unit circle.
  • The Unit Circle — The unit circle provides the extended definition of trigonometric ratios for all angles beyond 0°–90°, connecting the right-triangle ratios to coordinates in the plane.
  • Special Angle Values (30°, 45°, 60°) — Memorizing exact trig ratios for 30°, 45°, and 60° is essential for solving problems without a calculator and for understanding the unit circle.
  • Inverse Trigonometric Functions — Inverse trig functions (arcsin, arccos, arctan) reverse the ratios defined here, allowing you to recover an unknown angle when a ratio is known.
  • Pythagorean Theorem — The Pythagorean theorem (a² + b² = c²) is used to find the hypotenuse, which appears in the denominator of both sine and cosine, and underpins all right-triangle trigonometry.