Definition:Cosine
Definition
Definition from Triangle
In the above right triangle, we are concerned about the angle $\theta$.
The cosine of $\angle \theta$ is defined as being $\dfrac {\text{Adjacent}} {\text{Hypotenuse}}$.
Definition from Circle
Consider a unit circle $C$ whose center is at the origin of a cartesian plane.
Let $P = \tuple {x, y}$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis.
Let $AP$ be the perpendicular from $P$ to the $y$-axis.
Then the cosine of $\theta$ is defined as the length of $AP$.
Hence in the first quadrant, the cosine is positive.
Real Numbers
The real cosine function $\cos: \R \to \R$ is defined as:
| \(\ds \cos x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} + \cdots\) |
Complex Numbers
The complex cosine function $\cos: \C \to \C$ is defined as:
| \(\ds \cos z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots\) |
Also see
- Definition:Sine
- Definition:Tangent Function
- Definition:Cotangent
- Definition:Secant Function
- Definition:Cosecant
- Results about the cosine function can be found here.
Historical Note
The the symbol $\cos$ for cosine appears to have been invented by William Oughtred in his $1657$ work Trigonometrie, although some authors attribute it to Euler.
Linguistic Note
As with sine, the word cosine derives (erroneously, via a mistranslation) from the Latin sinus which (among other things) means fold, curve, winding or bay.
The co- prefix, as with similar trigonometric functions, is a reference to complementary angle: see Cosine of Complement equals Sine.
It is pronounced with the emphasis on the first syllable: co-sign.
$\cos x$ is voiced cosine (of) $x$, or (as written) cos $x$ (pronounced either coss or coz depending on preference).
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cosine
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cosine
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $5$: Eternal Triangles: Trigonometry
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): cosine