Review:-: Spherical Trigonometry

SPHERICAL TRIGONOMETRY

Review:Two rays that share the same endpoint form an angle. The point where the rays intersect is called the vertex of the angle. The two rays are called the sides of the angle. Angle Measurements (a) Sexagesimal System: In Sexagesimal System, an angle is measured in degrees, minutes and seconds. (b) Centesimal System: In Centesimal System, an angle is measured in grades, minutes and seconds. In this system, a right angle is divided into 100

(c) Circular System: In this System, an angle is measured in radians. In higher mathematics angles are usually measured in circular system. In this system a radian is considered as the unit for the measurement of angles. Definition of Radian: A radian is an angle subtended at the center of a circle by an arc whose length is equal to the radius.

acute angle

obtuse angle

right angle

straight angle

The sum of the measures of the angles of any triangle is 180°.

Right Triangle Trig Definitions B c a b

C • • • • • •

A

sin(A) = sine of A = opposite / hypotenuse = a/c cos(A) = cosine of A = adjacent / hypotenuse = b/c tan(A) = tangent of A = opposite / adjacent = a/b csc(A) = cosecant of A = hypotenuse / opposite = c/a sec(A) = secant of A = hypotenuse / adjacent = c/b cot(A) = cotangent of A = adjacent / opposite = b/a

Pythagoras Theorem states:

=

+

Basic Trigonometric Identities Quotient identities: Even/Odd identities:

tan( A) 

sin( A) cos( A)

cot( A) 

cos( A) sin( A)

cos( A)  cos(A)

sin(  A)   sin( A)

tan( A)   tan(A)

sec( A)  sec(A)

csc(  A)   csc( A) Odd functions

cot( A)   cot(A) Odd functions

Even functions

Reciprocal Identities:

1 sin( A ) 1 sin( A )  csc( A ) csc( A ) 

1 cos( A) 1 cos( A)  sec( A)

sec( A) 

1 tan( A) 1 tan( A)  cot( A)

cot( A) 

Pythagorean Identities:

sin 2 ( A)  cos 2 ( A)  1 tan 2 ( A)  1  sec2 ( A) 1  cot2 ( A)  csc2 ( A)

Quad I

Quad II

Quad III

cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0

cos(A)>0 sin(A)>0 tan(A)>0 sec(A)>0 csc(A)>0 cot(A)>0

cos(A)<0 sin(A)<0 tan(A)>0 sec(A)<0 csc(A)<0 cot(A)>0

cos(A)>0 sin(A)<0 tan(A)<0 sec(A)>0 csc(A)<0 cot(A)<0

Quad IV

Unit circle

• Radius of the circle is 1. • x = cos(θ) 1  cos( )  1 • y = sin(θ)  1  sin( )  1 • Pythagorean Theorem: x2  y 2  1 • This gives the identity: cos2 ( )  sin2 ( )  1 • Zeros of sin(θ) are where n is an integer. • Zeros of cos(θ) are where n is an integer.

n  2

 n

Trigonometric Identities Summation & Difference Formulas

sin( A  B)  sin( A) cos(B)  cos( A) sin(B) cos( A  B)  cos( A) cos(B)  sin( A) sin(B) tan( A  B) 

tan( A)  tan(B) 1  tan( A) tan(B)

Half Angle Formulas 1  cos(A)  A sin    2 2 1  cos(A)  A cos    2 2 1  cos(A)  A tan    1  cos(A) 2

Law of Sines & Law of Cosines Law of sines

sin( A) sin(B) sin(C )   a b c a b c   sin( A) sin(B) sin(C )

Law of cosines

c2  a2  b2  2abcos(C) b2  a2  c2  2accos(B) a2  b2  c2  2bccos(A)

SPHERICAL TRIGONOMETRY Spherical trigonometry is the study of curved triangles, triangles drawn on the surface of a sphere.

The diagram shows the spherical triangle with vertices A, B, and C. The angles at each vertex are denoted with Greek letters α, β, and γ. The arcs forming the sides of the triangle are labeled by the lower-case form of the letter labeling the opposite vertex.

On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180.

Definitions On a sphere, a great circle is the intersection of the sphere with a plane passing though the center, or origin, of the sphere Lune A lune is a part of the sphere which is captured between two great circles Spherical Triangle A spherical triangle is the intersection of three distinct lunes

Spherical Excess is the amount by which the sum of the angles (in the spherical plane only) exceed 180. Basic properties On the plane, the sum of the interior angles of any triangle is exactly 180°. On a sphere, however, the corresponding sum is always greater than 180° but also less than 540°. That is, 180° < α + β + γ < 540° in the diagram above. The positive quantity E = α + β + γ – 180° is called the spherical excess of the triangle. Since the sides of a spherical triangle are arcs, they can be described as angles, and so we have two kinds of angles: 1. The angles at the vertices of the triangle, formed by the great circles intersecting at the vertices and denoted by Greek letters. 2. The sides of the triangle, measured by the angle formed by the lines connecting the vertices to the center of the sphere and denoted by lower-case Roman letters. The second kind of angle is most interesting. In contrast to plane trigonometry, the sides of a spherical triangle are themselves are angles, and so we can take sines and cosines etc. of the sides as well as the vertex angles.

Spherical Coordinates is a coordinate system in three dimensions. The coordinate values stated below require r to be the length of the radius to the point P on the sphere. The value φ the angle between the z-axis, and the vector from the origin to point P, and θ the angle between the x-axis, and the same vector as in the figure. Then we can say the x; y; z coordinates are defined:

z =  cos 

r =  sin 

Polar triangle With the vertices of any spherical triangle ABC as poles let great circles be described, and let A' be that intersection of the circles described with B and C as poles which lies on the same side of BC as A does, and let B' and C' be similar intersections ; then the spherical triangle A'B'C' is called the polar triangle of ABC. Theorem. If one spherical triangle is the polar triangle of a second, then the second is also the polar triangle of the first.

Theorem. Let A'B'C' be the polar triangle of ABC ; then A = 180 - a', A' = 180 - a, B =.180 - b', B' = 180 - b, C - 180 - c', C' = 180 - c. RIGHT SPHERICAL TRIANGLES Relations between the sides and angles of a right spherical

Let ABC be a right spherical triangle, C the right angle, and a and I each less than 90. Draw the radii OA, OB, OC, thus forming the trihedral angle O-A nc. Take OP =OK =1. From A ' draw KD ⊥ OA , from P draw PE ⊥ OC, from E draw EA' ⊥ OA, and join A ' P.

Napier's rules 1. The sine of an angle is equal to the product of cosines of the opposite two angles. 2. The sine of an angle is equal to the product of tangents of the two adjacent angles.

Species.-Two angular quantities are said to be of the same species when they are both in the same quadrant, and of different species when they are in different quadrants. Since any or all the parts of a right spherical triangle may be less than or greater than 90°, it is necessary to have a method for the determination of the species of the parts. The following rules will be found to cover all cases: (1) An oblique angle and its opposite side are always of the same species. (2) If the hypotenuse is less than 90°, the two oblique angles and therefore the two sides of the triangle are of the same species; if the hypotenuse is greater than 90°, the two oblique

angles, and therefore the two sides, are of opposite species.

OBLIQUE SPHERICAL TRIANGLES Sine theorem (law of sines).-In any spherical triangle, the sines of the angles are proportional to the sines of the

opposite sides. Proof.-Let ABC be a spherical triangle. Construct the great circle arc CD, forming the two right spherical triangles CBD and CAD. Represent the arc CD by h.

Above law is useful in solving a spherical triangle when two angles and a side opposite one of them are given, or when two sides and an angle opposite one are given.

Cosine theorem (law of cosines).-In any spherical triangle, the cosine of any side is equal to the product of the

cosines of the two other sides, increased by the product of the sines of these sides times the cosine of their included angle. Proof.-Let ABC be a spherical triangle cut from the surface of a sphere, with center 0, and radius OA chosen as unity. At any point D in OA, draw a plane EDF perpendicular to the edge OA and meeting the faces of the trihedral angle in DE, DF,

In the figure, both b and c are less than 90°, while no restriction is placed upon α or a. The resulting formulas are true, however, in general, as may easily be shown. In the figure, let ABC be a spherical triangle with c > 90° and b < 90°. Complete the great circle arcs to form the triangle DCA, in which AD= (180° - c) < 90°. The parts of DCA are 180° - c, 180° - α,180° - a,β, and b. Then cos(180° -a) = cos b cas (180° -c) +sin b sin (180° c)cos(180° - α). . . . cas a = cas b cos c + sin b sin c cos α.

Theorem.-The cosine of any angle of a spherical triangle is equal to the product of the sines of the two other angles multiplied by the cosine of their included side, diminished by the product of the cosines of the two other angles. Let ABG be the spherical triangle of which A' B'G' is the polar triangle. Then a = 180° - α.',b = 180° - β', c = 180° - γ', and a = 180° - a'.

Given the three sides to find the angles Let ABC be a spherical triangle with given sides a, b, and c

Given the three angles to find the sides

Rules for species in oblique spherical triangles If a side (or an angle) differs from 90° by a larger number of degrees than another side (or angle) in the triangle, it is of the same species as its opposite angle (or side).

Cases.-In the solution of oblique spherical triangles, the six following cases arise: CASE I. Given the three sides. Example.-Given a= 46° 20' 45", b = 65° 18' 15", c = 90° 31' 46";

to find α,β,γ. Case II. Given the three angles to find the three sides

Case III. Given two sides and the included angle

Given a= 103° 44.7', b = 64° 12.3', γ = 98° 33.8'; Case IV. Given two angles and the included side Given α = 59° 4' 25", β = 88° 12' 24", C = 47° 42' I"; Case V. Given two sides and the angle opposite one of them Given a = 46° 20' 45", b = 65° 18' 15", IX = 40° 10' 30" Case VI. Given two angles and the side opposite one of them

Given IX = 29° 2' 55", β = 45° 44' 6", a = 35° 37' 18"

Area of a spherical triangle.

L'Huilier's formula.-This is a formula for determining the spherical excess directly in terms of the .sides. It may be derived as follows:

APPLICATIONS OF SPHERICAL TRIGONOMETRY Definitions and notations.-In all the applications of spherical trigonometry to the measurements of arcs of great circles on the surface of the earth, and to problems of astronomy. A meridian is a great circle of the earth drawn through the poles N and S. The meridian NGS passing through Greenwich, England, is called the principal meridian. The diameter of the Earth is assumed to be 12,756 km The longitude of any point P on the earth's surface is the angle between the principal meridian NGS and the meridian W NPS through P. It is measured by the great circle arc, CA, of the equator between the points where the meridians cut the equator. If a point on the surface of the earth is west of the principal meridian, its longitude is positive. If east, it is negative. A point 70° west of the principal meridian is usually designated as in "longitude 70° W." "Longitude 70° E." means in 70° east longitude. The letter Θ is used to designate longitude. The latitude of a point on the surface of the earth is the number of degrees it is north or south of the equator, measured along a meridian. Latitude is positive when measured north of the equator, and negative when south. The letter ɸ is used to designate latitude. Loxodrome or Rhumb Line The geometric representation of the curve on the spherical surface of the earth that intersects all meridians at the same angle

The terrestrial triangle. Given two points on the earth's surface, H with latitude ɸ1 longitude Θl, and P with latitude ɸ2 longitude Θ2; then the arcs HN= 90 - ɸ1, PN= 90 - ɸ2,and HP, which represents the distance between the points, form a spherical triangle called the terrestrial triangle. The angle HNP= Θ2- Θ1 The angle NHP is the bearing of P from H, and angle NPH the bearing of H from P. The bearing will be represented by γ The bearing of a line is usually the smallest angle which the line or path makes with the meridian through the point. Find the shortest distance, between New York, 40° 45' N., 73° 58' W., and Chicago, 41° 50' N., 87° 35' W.