Angle

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This article is about angles in geometry. For other articles, see Angle (disambiguation)

An Angle (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Greek Template:Polytonic (angulοs) crooked, curved; both connected with the Aryan or Indo-European root ank-, to bend) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles provide a means of expressing the difference in slope between two rays meeting at a vertex without the need to explicitly define the slopes of the two rays. Angles are studied in geometry and trigonometry.

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.

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Units of measure for angles

In order to measure an angle, a circle centered at the vertex is drawn. Since the circumference of a circle is always directly proportional to the length of its radius, the measure of the angle is independent of the size of the circle. Note that angles are dimensionless, since they are defined as the ratio of lengths.

  • The radian measure of the angle is the length of the arc cut out by the angle, divided by the circle's radius. The SI system of units uses radians as the (derived) unit for angles. This is also roughly subdivided into the mil, which has several definitions in practice. Because of the relationship to arc length, radians are a special unit. Sines and cosines whose argument is in radians have particular analytic properties, just as do exponential functions in the base e. (As we've discovered, this is no coincidence).
  • The degree measure of the angle is the length of the arc, divided by the circumference of the circle, and multiplied by 360. The symbol for degrees is a small superscript circle, as in 360°. 2π radians is equal to 360° (a full circle), so one radian is about 57° and one degree is π/180 radians. Degrees are further broken down into minutes of arc and seconds of arc, which are 1/60th and 1/3600th of a degree, respectively. Minutes of arc are commonly encountered in discussions of external ballistics, as a minute of arc covers almost exactly 1 inch at 100 yards (1 m at 1200 m). A rifle capable of shooting "1 MOA", one minute of arc, can place all shots within 1 inch at 100 yards, 2 inches at 200 yards, etc. Minutes of arc are also used in navigation: a nautical mile was historically defined as a minute of arc along a great circle of the Earth.
  • The grad, also called grade, gradian or gon, is an angular measure where the arc is divided by the circumference, and multiplied by 400. It is used mostly in triangulation.
  • The point is used in navigation, and is defined as 1/32 of a circle, or exactly 11.25°.
  • The full circle or full turns represents the number or fraction of complete full turns. For example, π/2 radians = 90° = 1/4 full circle
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Conventions on measurement

A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 is north-east. Negative bearings are not used in navigation, so north-west is 315.

In mathematics radians are assumed unless specified otherwise because this removes the arbitrariness of the number 360 in the degree system and because the trigonometric functions can be developed into particularly simple Taylor series if their arguments are specified in radians.

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Types of angles

An angle of π/2 radians or 90°, one-quarter of the full circle is called a right angle.

Two line segments, rays, or lines (or any combination) which form a right angle are said to be either perpendicular or orthogonal:

Image:Right angle.svg Image:Angle obtuse acute straight.svg
  • Angles smaller than a right angle are called acute angles (less than 90 degrees)
  • Angles larger than a right angle are called obtuse angles (more than 90 degrees, less that 180).
  • Angles equal to two right angles are called straight angles (equal to 180 degrees).
  • Angles larger than two right angles are called reflex angles (more than 180 degrees).
  • The difference between an acute angle and a right angle is termed the complement of the angle
  • The difference between an angle and two right angles is termed the supplement of the angle.
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Some facts

In Euclidean geometry, the inner angles of a triangle add up to π radians or 180°; the inner angles of a quadrilateral add up to 2π radians or 360°. In general, the inner angles of a simple polygon with n sides add up to (n − 2) ×   π radians or (n − 2)  ×  180°.

If two straight lines intersect, four angles are formed. Each one has an equal measure to the angle across from it; these congruent angles are called vertical angles.

If a straight transversal line intersects two parallel lines, corresponding (alternate) angles at the two points of intersection are equal; adjacent angles are supplementary, that is they add to π radians or 180°.



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A formal definition

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Using trigonometric functions

A Euclidean angle is completely determined by the corresponding right triangle. In particular, if <math>\theta</math> is a Euclidean angle, it is true that

<math>\cos \theta = \frac{x}{\sqrt{x^2 + y^2}}</math>

and

<math>\sin \theta = \frac{y}{\sqrt{x^2 + y^2}}</math>

for two numbers <math>x</math> and <math>y</math>. So an angle can be legitimately given by two numbers <math>x</math> and <math>y</math>.

To the ratio <math>\frac{y}{x}</math> there correspond two angles in the geometric range <math>0 < \theta < 2\pi </math>, since

<math>\frac{\sin \theta }{\cos \theta } = \frac{\frac{y}{\sqrt{x^2 + y^2}}}{\frac{x}{\sqrt{x^2 + y^2}}} = \frac{y}{x} = \frac{-y}{-x} = \frac{\sin (\theta + \pi)}{\cos (\theta + \pi) } </math>
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Using rotations

Suppose we have two unit vectors <math>\vec{u}</math> and <math>\vec{v}</math> in the euclidean plane <math>\mathbb{R}^2</math>. Then there exists one linear application (a rotation), and one only, from <math>\mathbb{R}^2</math> to <math>\mathbb{R}^2</math> that maps <math>u</math> onto <math>v</math>. Let r be such a rotation. Then the relation <math>\vec{a}\mathcal{R}\vec{b}</math> defined by <math>\vec{b}=r(\vec{a})</math> is an equivalence relation and we call angle of the rotation r the equivalence class <math>\mathbb{T}/\mathcal{R}</math>, where <math>\mathbb{T}</math> denotes the unit circle of <math>\mathbb{R}^2</math>. The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector <math>(1,0)</math>, then for any point M on <math>\mathbb{T}</math> at distance <math>\theta</math> from <math>(1,0)</math> (on the circle), let <math>\vec{u}=\overrightarrow{OM}</math>. If we call <math>r_\theta</math> the rotation that transforms <math>(1,0)</math> into <math>\vec{u}</math>, then <math>\left[r_\theta\right]\mapsto\theta</math> is a bijection, which means we can identify any angle with a number between 0 and <math>2\pi</math>.

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Angles in different contexts

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula

<math>\mathbf{u} \cdot \mathbf{v} = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.</math>

This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>.

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.

Two intersecting planes form an angle, called their dihedral angle. It is defined as the acute angle between two lines normal to the planes.

Also a plane and an intersecting line form an angle. This angle is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.

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Angles in Riemannian geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G,

<math>

\cos \theta = \frac{g_{ij}U^iV^j} {\sqrt{ \left| g_{ij}U^iU^j \right| \left| g_{ij}V^iV^j \right|}}. </math>

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Angles in astronomy

In astronomy, one can measure the angular separation of two stars by imagining two lines through the Earth, each one intersecting one of the stars. Then the angle between those lines can be measured; this is the angular separation between the two stars.

Astronomers also measure the apparent size of objects. For example, the full moon has an angular measurement of approximately 0.5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

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Angles in maritime navigation

The modern format of angle used to indicate longitude or latitude is hemisphere degree minute.decimal, where there are 60 minutes in a degree, for instance N 51 23.438 or E 090 58.928.

The obsolete (but still commonly used) format of angle used to indicate longitude or latitude is hemisphere degree minute' second", where there are 60 minutes in a degree and 60 seconds in a minute, for instance N 51 23′26″ or E 090 58′57″

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See also

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External links

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