SPHERICAL TRIGONOMETRY
DEFINITION OF TERMS
The sphere is the set of all points in a three-dimensional space such that the distance of each from a fixed point is constant. The fixed point and the given distance are called the center and the radius of the sphere respectively.

The intersection of a plane with a sphere is a circle. If the plane passes through the center of the sphere, the intersection is a great circle; otherwise, the intersection is a small circle.
A line perpendicular to the plane of a circle and through the center of the sphere is called the axis of the circle. The intersection of this axis and the sphere are called the poles of the circle. Opposite ends of a diameter are identified as antipodal points. Two great circles intersecting in a pair of antipodal points divide the sphere into four regions called lunes. Thus a lune is bounded by the arcs of two great circles.

The polar distance (in angular units) of a circle is the least distance of a point on the circle to its pole.
Two distinct points on the sphere which are not ends of a diameter divide the great circle into two arcs. The shorter arc is called the minor arc.

SPHERICAL TRIANGLES
A spherical triangle is that part of the surface of a sphere bounded by three arcs of great circles. The bounding arcs are called the sides of the spherical triangle and the intersections of these arcs are called the vertices of the spherical triangle. The angle formed by two intersecting arcs is called a spherical angle. Like a plane triangle, the spherical triangle has also six parts – three angles and three sides. The sides a, b, and c are measured by the corresponding faces of the trihedral angle. Important Propositions from Solid Geometry:

1. If two sides are equal, the angles opposite are equal and conversely. 2. If two sides are unequal, the angles opposite are unequal and the greater side is opposite the greater angle and conversely. 3. The sum of any two sides is...

...ENGTRIG: LECTURE # 4.2 SphericalTrigonometrySphericalTrigonometry
Engr. Christian Pangilinan
Areas of a Spherical Triangle
A=
π R2 E
180o E R
E = A + B + C − 180o
Where:
spherical excess radius of the sphere
Spherical Triangles Part of the surface of the sphere bounded by three arcs of three great circles Right Spherical Triangle – aspherical triangle containing at least one right angle
If the sides are known instead of the angles, then L’Huiller’s Formula can be used to solve for the spherical excess
1 s s − a s − b s−c tan E = tan tan tan tan 2 2 2 2 2 a+b+c Where: s= 2
Solutions to Right Spherical Triangles (C = 90o)
Sides a, b, c are based on the corresponding arc lengths S = rθ that is based on its corresponding interior angles S a = rθ BOC ; Sb = rθ AOC ; Sc = rθ AOB and that a + b + c < 360o Angles A, B, C of a spherical triangle are measured by the corresponding dihedral angles of the trihedral angle A : B − OA − C ; B : A − OB − C and
C : A − OC − B
Napier’s Rules: 1. The sine of any middle part is equal to the product of the tangents of the adjacent parts 2. The sine of any middle part is equal to the product of the opposite parts *“co” indicates complement
s i n ( middle ) = product of t a n ( adjacent ) s i n (...

...SphericaltrigonometrySphericaltrigonometry is that branch of spherical geometry which 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. Sphericaltrigonometry is of great importance for calculations in astronomy, geodesy and navigation.
The origins of sphericaltrigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Todhunter's text book Sphericaltrigonometry for the use of colleges and Schools. This book is now readily available on the web.[1] The only significant developments since then have been the application of vector methods for the derivation of the theorems and the use of computers to carry through lengthy calculations.
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Preliminaries
Eight spherical triangles defined by the intersection of three...

...Early trigonometry
The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. But pre-Hellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were studied instead, a field that would be better called "trilaterometry".[6]The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere.[2] Based on one interpretation of the Plimpton 322 cuneiform tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants.[7] There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table.The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC.[2] The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes (c. 1680–1620 BC), contains the following problem related to trigonometry:[2]"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the...

...Trigonometry (from Greek trigōnon "triangle" + metron "measure"[1]) is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.[2] It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Sphericaltrigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomy and navigation. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry.
Contents
f one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely...

...The Way Trigonometry is used in Astronomy
By: Joanna Matthews
Practical Applications of Advanced Mathematics
Mrs. Amy Goodrum
July 15, 2003
Abstract
This report is about how trigonometry is used in Astronomy. Even though trigonometry is applied in many areas, such as engineering, chemistry, surveying, and physics, it is mainly used in astronomy Trigonometry is used to find the distance of stars, the distance from one planet to another and from one plant to the sun. It is possible to find the radius of the Earth also. This report will basically give more insight in the way trigonometry and astronomy goes hand in hand.
Background
Trigonometry comes from a Greek word "trigonometria" put together from these 3 words: Tri (three) gonia (angle) metro (measure). Trigonometry has been around for many centuries, but in 140 BC a man named Hipparchus apparently wrote 12 books on the table of chords and became the founder of trigonometry. He was the first Greek mathematician to study triangular geometry. This study led him to write the 12 books.
Other people have added to Hipparchus’ work, but the two people that stands out the most are were Menelaus (ca. AD 100) and Ptolemy (ca. AD 100). Menelaus was a Greek mathematician that created six books on tables of chords. He created a couple of triangle properties. Menelaus had a big hand in...

...MA1310 College Mathematics II LESSON PLAN WEEK 5
Dr. Tonjes September 2011
LESSON: Oblique Triangles, Laws of Sines and Cosines
INTRODUCTION:
Student will demonstrate how to apply laws of sines and cosines to oblique triangles.
OBJECTIVES:
After completing this unit, the student will be able to:
6. Use the Law of Sines and the Law of Cosines to solve oblique triangle problems.
6.1. Summarize the Law of Sines.
6.2. Find the area of an oblique triangle using the sine function.
6.3. Judge when an ambiguous case of the Law of Sines occurs.
6.4. Solve applied problems using the Law of Sines.
6.5. Summarize the Law of Cosines.
6.6. Use the Law of Cosines to solve oblique triangle problems.
6.7. Solve applied problems using the Law of Cosines.
6.8. Find the area of an oblique triangle using Heron’s formula.
PROCEDURE:
Content
Activity
Objectives
Present objectives and purpose of lesson.
Law of Sines and Law of Cosines
Generalize the sine and cosine relations of the right triangles to oblique triangles by defining the two laws.
Law of Sines: This law relates the three sides of any triangle to the angles opposite the sides, typically labeled a, b, and c for the sides and A, B, and C for the angles.
Law of Cosines: This law relates one side to the other two sides and its corresponding angle:
Relate that either of these relations reduce to simpler forms for the case of right triangles, particularly:
1. Law of Sines reduces to c = c = c.
2. Law of...

...CHAPTER 6 CIRCULAR FUNCTIONS AND TRIGONOMETRY
CONTENTS
-Angles and Their Measures
-Degrees and Radians
-Angles in Standard Position and Coterminal Angles
-Angles in a Quadrant
-The Unit Circle
-Coordinates of Points on the Unit Circle
-The Sine and Cosine Function
-Values of Sine and Cosine Functions
-Graphs of Sine and Cosine Functions
-The Tangent Function
-Graph of Tangent Function
-Trigonometric Identities
-Sum and Difference of Formulas for Sine and Cosine
-Trigonometric Functions of an angle
-Values of the Function of an Angle
-Simple Trigonometric Equations
-Right Triangle Trigonometry
-Angle of Elevation/Depression
- Solving Right Triangles
-The Law of Sines
-The Law of Cosines
CERAE
CHAPTER 6 CIRCULAR FUNCTIONS AND TRIGONOMETRY
EXPERIENCE
What I Have Learned?
I learned many things in trigonometry especially in chapter 6 , I learned many lessons specifically in the lessons 6.1-6.4 (Angles and Their Measures, Degrees and Radians, Angles in Standard Position and Coterminal Angles, Angles in a Quadrant, The Unit Circle, Coordinates of Points on the Unit Circle, The Sine and Cosine Function , Values of Sine and Cosine Functions, Graphs of Sine and Cosine Functions, and The Tangent Function).
How Did I learn?
I learned many things in Chapter 6 but I couldn’t learned or understand all these things, topics or lessons by myself, I learned all these topics of course through our...

...Teaching trigonometry using Empirical Modelling
0303417
Abstract
The trigonometric functions sin(x), cos(x) and tan(x) are relationships that exist between the angles
and length of sides in a right-angled triangle. In Empirical Modelling terms, the angles in a triangle
and the length of the sides are observables, and the functions that connect them are the definitions.
These well-defined geometric relationships can be useful when teaching GCSE-level students about
the functions, as they provide a way to visualise what can be thought of as fairly abstract functions.
This paper looks at how different learning styles apply to Empirical Modelling, and presents a practical example of their use in a model to teach trigonometry.
1 Introduction
The trigonometric functions sin(x), cos(x) and tan(x)
are relationships that exist between the angles and
length of sides in a right-angled triangle. In Empirical Modelling terms, the angles in a triangle and the
length of the sides are observables, and the functions
that connect them are the definitions. These welldefined geometric relationships can be useful when
teaching GCSE-level students about the functions,
as they provide a way to visualise what can be
thought of as fairly abstract functions. Rather than
teaching students by showing them diagrams in an
instructive way (already a good way of doing it), a
constructive approach may allow students to gain a
better understanding...