Angle : Angular measurement

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	Angle : Angular measurement This article is about angles in geometry. See: fly fishing for the technique of using a bait[?] and hook to catch fish. Angle tribe for the Germanic tribe that moved to Britain. Angle, Pembrokeshire[?] for the place in Wales. An angle is the figure formed by two line segments extending from a point, the vertex of the angle. Angles are studied in geometry and trigonometry. 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 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.

Measuring angles

In order to measure an angle, a circle centered at the vertex is drawn. The radian measure of the angle is the length of the arc cut out by the angle, divided by the circle's radius. 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°. The grad, also called grade or gon, is a angular measure where the arc is divided by the circumference, and multiplied by 400. It is used mostly in triangulation.

2π radians is equal to 360° (a full circle), so one radian is about 57° and one degree is π/180 radians.

Mathematicians generally prefer angle measurements in radians 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. The SI system of units uses radians as the (derived) unit for angles.

Types of angle

An angle of π/2 radians or 90 degrees, one-quarter of the full circle is called a right angle. Two line segments which form a right angle are said to be perpendicular:

Angles smaller than a right angle are called acute; angles larger than a right angle are called obtuse. Angles larger than two right angles are called reflex angles.

Angles in different contexts

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

<math>\mathbf{u} \cdot \mathbf{v} = \cos(\theta) ||\mathbf{u}|| \cdot ||\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 of two intersecting curves is defined to be the angle between the tangents at the point of intersection.

See also solid angle for a concept of angle in three dimensions.

Angles in Riemannian Geometry

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where <math>U</math> and <math>V</math> are tangent vectors and <math>g_{ij}</math> are the components of the metric tensor <math>G</math>,

<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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