Definition of Napier's rule. : either of two rules in spherical trigonometry: the sine of any part is equal to the product of the tangents of the adjacent parts and the sine of any part is equal to the product of the cosines of the opposite parts.
Likewise, people ask, what is Napier's formula?
Then Napier has two rules: The sine of a part is equal to the product of the tangents of the two adjacent parts. The sine of a part is equal to the product of the cosines of the two opposite parts.
Subsequently, question is, what is right spherical triangle? A right spherical triangle, on the other hand, is a spherical triangle whose one of its angles measures 90°. Spherical triangles are labeled with angles A, B and C, and respective sides a, b, and c opposite these angles. For right spherical triangles, it is customary to set C = 90°.
Additionally, what is Napier's analogy?
Definition of Napier's analogies. : four formulas giving the tangent of half the sum or difference of two of the angles or sides of a spherical triangle in terms of the others.
How many degrees are in a spherical triangle?
What are the properties of spherical triangle?
The three angles of a spherical triangle must together be more than 180° and less than 540° . 7. The greater side is opposite the greater angle , if tow sides are equal their opposite angles are equal . , and if one side of the triangle 90° it is called a quadrantal triangle .
What is oblique spherical triangle?
Oblique Spherical Triangle. Definition of oblique spherical triangle. Spherical triangles are said to be oblique if none of its included angle is 90° or two or three of its included angles are 90°. Spherical triangle with only one included angle equal to 90° is a right triangle.
What is the law of tangents used for?
Application. The law of tangents can be used to compute the missing side and angles of a triangle in which two sides a and b and the enclosed angle γ are given. From. one can compute α − β; together with α + β = 180° − γ this yields α and β; the remaining side c can then be computed using the law of sines.
What is Polar triangle?
Definition of polar triangle. : a spherical triangle formed by the arcs of three great circles each of whose poles is the vertex of a given spherical triangle.
How do you define angles?
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane.
How do you find the area of a spherical triangle?
Area. Let R be the radius of the sphere on which a triangle resides. If angles are measured in radians, the area of a triangle is simply R2E where E is the spherical excess, defined above. In degrees the formula for area is πR2E/180.
What is a triangle on a sphere called?
A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar triangle, and is sometimes called an Euler triangle (Harris and Stocker 1998).
What is spherical trigonometry used for?
Spherical trigonometry is the branch of spherical geometry that deals with the relationships between trigonometric functions of the sides and angles of the spherical polygons (especially spherical triangles) defined by a number of intersecting great circles on the sphere.
What is spherical excess?
Definition of spherical excess. : the amount by which the sum of the three angles of a spherical triangle exceeds two right angles.
Who discovered spherical trigonometry?
Why is spherical geometry important?
Spherical geometry is important in navigation, because the shortest distance between two points on a sphere is the path along a great circle. Riemannian Postulate: Given a line and a point not on the line, every line passing though the point intersects the line.
How do you find the angle measure of a sphere?
Area and angles on a sphere
- If A, B, and C are the interior angles of a triangle on a unit sphere, the excess (A+B+C) – Pi is exactly equal to the area of the triangle.
- The area of a sector with vertex angle A (measured in radians) is 4 Pi * A / 2 Pi = 2 A.
- The defect of any polygonal figure on a unit sphere is equal to its area.
What makes up a sphere?
A sphere is a geometrical figure that is perfectly round, 3-dimensional and circular – like a ball. Geometrically, a sphere is defined as the set of all points equidistant from a single point in space.
How many angles are in a sphere?
There are essentially 360 o of 360 o in a sphere. Edit: maybe 180 not that i think about it. A circle would only have to make a half turn. Degrees are used to measure in two dimensions.
Can a triangle have 270 degrees?
A 270–degree triangle is possible when it is not limited to two dimensions.
What is the maximum angle of a triangle?
Each triangle has an angle sum of 180 degrees. Therefore the total angle sum of the quadrilateral is 360 degrees.
How is spherical geometry used in real life?
Spherical Geometry is also known as hyperbolic geometry and has many real world applications. One of the most used geometry is Spherical Geometry which describes the surface of a sphere. Spherical Geometry is used by pilots and ship captains as they navigate around the world.
What is a line segment on a sphere?
Lines. As in plane geometry, a line–segment on a sphere is the shortest ‘line‘ connecting two points – over the surface of the sphere. So the maximum ‘extension' of a line–segment (and therefore our line) is a circle, with the same radius as the sphere.
Can triangles have more than 180 degrees?
It is no longer true that the sum of the angles of a triangle is always 180 degrees. Very small triangles will have angles summing to only a little more than 180 degrees (because, from the perspective of a very small triangle, the surface of a sphere is nearly flat).