1 CHAPTER 3 PLANE AND SPHERICAL TRIGONOMETRY 3.1 Introduction It is assumed in this chapter that readers are familiar with the usual elementary formulas encountered in introductory trigonometry. We start the chapter with a brief review of the solution of a plane triangle. While most of this will be familiar to readers, it is suggested that it be not skipped over entirely, because the examples in it contain some cautionary notes concerning hidden pitfalls. This is followed by a quick review of spherical coordinates and direction cosines in threedimensional geometry. The formulas for the velocity and acceleration components in twodimensional polar coordinates and three-dimensional spherical coordinates are developed in section 3.4. Section 3.5 deals with the trigonometric formulas for solving spherical triangles. This is a fairly long section, and it will be essential reading for those who are contemplating making a start on celestial mechanics. Section 3.6 deals with the rotation of axes in two and three dimensions, including Eulerian angles and the rotation matrix of direction cosines. Finally, in section 3.7, a number of commonly encountered trigonometric formulas are gathered for reference. 3.2 Plane Triangles. This section is to serve as a brief reminder of how to solve a plane triangle. While there may be a temptation to pass rapidly over this section, it does contain a warning that will become even more pertinent in the section on spherical triangles. Conventionally, a plane triangle is described by its three angles A, B, C and three sides a, b, c, with a being opposite to A, b opposite to B, and c opposite to C. See figure III.1.

B FIGURE III.1 c C b A a

2

It is assumed that the reader is familiar with the sine and cosine formulas for the solution of the triangle: a b c = = sin A sin B sin C

3.2.1

and

a 2 = b 2 + c 2 − 2bc cos A,

3.2.2

and understands that the art of solving a triangle involves recognition as to which formula is appropriate under which circumstances. Two quick examples - each with a warning - will suffice. Example: A plane triangle has sides a = 7 inches, b = 4 inches and angle B = 28o. Find the angle A.

See figure III.2. We use the sine formula, to obtain

sin A =

7 sin 28o = 0.821575 4

A = 55o 14'.6

3 The pitfall is that there are two values of A between 0o and 180o that satisfy sin A = 0.821575, namely 55o 14'.6 and 124o 45'.4. Figure III.3 shows that, given the original data, either of these is a valid solution.

The lesson to be learned from this is that all inverse trigonometric functions (sin-1 , cos-1 , tan-1 ) have two solutions between 0o and 360o . The function sin-1 is particularly troublesome since, for positive arguments, it has two solutions between 0o and 180o . The reader must always be on guard for "quadrant problems" (i.e. determining which quadrant the desired solution belongs to) and is warned that, unless particular care is taken in programming calculators or computers, quadrant problems are among the most frequent problems in trigonometry, and especially in spherical astronomy. Example: Find x in the triangle illustrated in figure III.4.

4 Application of the cosine rule results in 25 = x2 + 64 − 16x cos 32o Solution of the quadratic equation yields x = 4.133 or 9.435 This illustrates that the problem of "two solutions" is not confined to angles alone. Figure III.4 is drawn to scale for one of the solutions; the reader should draw the second solution to see how it is that two solutions are possible. The reader is now invited to try the following "guaranteed all different" problems by hand calculator. Some may have two real solutions. Some may have none. The reader should draw the triangles accurately, especially those that have two solutions or no solutions. It is important to develop a clear geometric understanding of trigonometric problems, and not merely to rely on the automatic calculations of a machine. Developing these critical...

...SPHERICALTRIGONOMETRY
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...

...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.
Comment: this article is best viewed with Preference->Appearance->mathJax.
Preliminaries
Eight spherical triangles defined by the intersection of three...

...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 (...

...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...

...To my perpetual knowledge, nobody has ever succeeded when challenging the onboard facilities and staff of an Aeroplane. The only possible reason for this is that David Cameroon assassinates anyone who even thinks about with a toothpick, I found out the hard way... Aeroplanes, from a distance, look like beautiful, massive and inviting, winged creatures, inside? They are far from it. Aeroplane architects spend so much time on the outer decor of the plane; they hardly remember there’s an inside as well! If the plane is colourful and has big ‘jets’ then I guess nothing else matters, Right? Right? Wrong!
Let’s start off with the exquisite, elegant and engaging stewardesses who for some reason feel it’s necessary to give me directions to my seat. I managed to put my trousers on and dress myself accordingly this morning but I need some stranger to continuously shout random letters and numbers at me. B29? STRAIGHT! G16? STRAIGHT! Dwelling more into the predicament of Stewardesses, one thing I hate about them is the fact that they always smile. They come into work smiling, and even after a hard day’s work, they leave smiling... So to prove this little theory of mine I decided to do an ingenious, harebrained experiment, want to hear it? No? Ok here it is...
About a month ago I was on a trip to Hawaii with my Mother-In-Law, when I decided to stand up from my seat (whilst the seat belt sign was flashing, yes ... I know, rebellious I Am.) and slap a...

...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...

...DIVERSITY IN LIVING ORGANISMS
1. What is classification?
The method of arranging the organism into different groups on the basis of similarities and dissimilarities is called classification.
The science of classification is called taxonomy or systematics
The father of taxonomy is Carl Linnaeus.
2. Why do we classify organisms? or What is the importance of classification?
• Classification makes the study of the variety of organisms existing in the living world, convenient, easy and time saving.
• Classification projects before us a picture of all life forms at a glance.
• Classification helps us in exploring the diversity of life forms.
• Classification helps us to study about the interrelationship between different groups of organisms and in turn provides us information about their evolution.
• It serves as a base for the development of other biological sciences such as ecology, behavioral sciences, biogeography evolution etc.
3. What is biodiversity?
The diversity of life forms found in a particular region is called its biodiversity.
4. What do you think is a more basic characteristic for classifying organisms?
The kind of cells they are made up of is a more basic characteristic for classifying organisms. If organisms are classified on the basis of kinds of cells, they are made of – there are two groups of organisms - prokaryotic and eukaryotic.
5. What is the primary...

...~TUDENTIDNO
FIRST TRIMESTER EXAMINATION, ZOO+2005 SESSION
PMT 0045 - IT MATHEMATICS 1
( Alpha Information Technology studdnts only )
17 AUGUST 2004
9.00 a.m - 11 .OO a.m
( 2 Hours )
INSTRUCTIONS TO STUDENT
1. This Question paper consists of 4 pages excluding the qover page and formula sheet.
2. Answer FOUR out of FIVE questions.
3. Please print all your answers in the Answer Booklet prcivided. All necessary working
MUST be shown.
Instruction: Answer FOUR questions only.
1 uestion
O
(a) Verify the identity
:
(b) Find the exact value of
cscB-sinB =cotBcosB.
.
(
3 marks )
(
3 marks )
(c) If cosA =f with A in Quadrant I and cosB = i with B in Quadrant IV, find the
exact value of
0
cos(A + B)
ii)
sin 23
iii)
sin2
A
( 9 marks )
(d) Solve the equation 2 cos2 x - 1 lcosx = -5,
0 < x c 27~
( 5 marks )
Continued.......
NilLCY/GWW
l/4
17 AUGUST 2004
IT MATHEMATICS I
PMT 0045
Ouestion 2:
(a) Solve A ABC, given a = 22 meters, b = 12 meters and A = 41’. Find the remaining
side and angles by using Law of Sines. Give your answer id two decimal places.
( 7 marks )
(b) Find the remaining angles and side of triangle below. Give tour answer in two
decimal places.
( 6 marks )
ii)
Convert from polar to rectangular coordinate for
( 7 marks )
Continued..
NIILCYIGWW
214
I
.....
Question 3:
(a) Find the equation of the parabola with focus at (2,-5)...