A C

B

Introduction To Spherical Trigonometry/Astronomy

Fred Sawyer Chicago 2005

B

A C

A

B

C

a

b

c

Rule 1.

The sum of the lengths of a spherical triangle's sides is always less than 360º.

A

B

C

a

b

c

Rule 2.

The sum of the angles at its vertices is greater than 180º and less than 540º.

A

B

C

a

b

c

Rule 3.

The sum of the lengths of any two sides is greater than the length of the third side.

A

B

C

a

b

c

Rule 4.

If a side (or angle) differs from 90º by more than another side (or angle), then it is in the same quadrant as its opposite angle (or side).

In other words, they are either both greater than 90º or both less than 90º.

A

B

C

a

b

c

Rule 5.

Half the sum of two sides of a spherical triangle must be in the same quadrant as half the sum of the two opposite angles.

A

B

C

a

b

c

Cosine Law

The Fundamental LawOf Spherical Trigonometry

cos cos cos sin sin cosc a b a b C

A

B

C

a

b

c

Cosine Law

The Fundamental LawOf Spherical Trigonometry

cos cos cos sin sin cosc a b a b C Acbcba cossinsincoscoscos

cos cos cos sin sin cosb a c a c B

A

B

C

a

b

c

Cosine Law

The Fundamental LawOf Spherical Trigonometry

cBABAC cossinsincoscoscos

A

B

C

a

b

c

Sine Law

cCbBaA sin/sinsin/sinsin/sin

A

B

C

a

b

c

cos cos sin cot sin cota C a b C B

Four Element Equation

A

B

C

a

b

c

Given any 3 elements of a spherical triangle, it is possible to solve for the other 3.

A

B

C

a

b

c

cos cos cos sin sin cosc a b a b C

sin / sin sin / sin sin / sina A b B c C

B

C

a

b

cos cos sin cot sin cota C a b C B

Four Element Equation

A

a

b

B

aAbB sin/sinsinsin

sin / sin sin / sin sin / sina A b B c C

But… Is B acute or obtuse??? Appeal to Rules 4 and 5

Most Difficult Case

A

B

a

b

Acbcba cossinsincoscoscos

cos cos cos sin sin cosb a c a c B

c

A

B

a

b

AabBba

AbbBaac

coscossincoscossin

coscossincoscossincos

c

P

Z

S

az

t

Given latitude, declination, and hour angle –

Find solar altitude.

P

Z

S

az

t

tcossinsincoscoscos

tcoscoscossinsinsin

P

Z

S

az

t

Given altitude, azimuth and latitude,

Find the solar declination and hour angle.

P

Z

S

az

t

Cosine law gives us declination from altitude, azimuth and latitude.

Then sine law gives us hour angle from declination, azimuth and altitude.

P

Z

S

az

t

Alternatively, use the Four Element equation to obtain the hour angle directly from altitude, azimuth and latitude.

Then use the sine law or the cosine law to find the declination.

P

Z

S

az

t

Given latitude, hour angle, and declination,

Find the sun’s azimuth.

P

Z

S

az

t

aztt cotsincotsincoscos

P

Z

S

az

t

t

taz

sin

cotsincoscoscot

t

taz

sin

tancoscossincot

Z

S

P

90

t

H

tcot90sincotsin90coscos tcotcotsin ttansintan

P

Z

S

P

Z

S

Prosthaphaeretical Arc

Hectemoral Arc (complement)

t

Hec

sin

sinsin

Hect

Reducing A PlaneTo The EquivalentHorizontal

Begin with a horizontal plane at latitude Spike the celestial sphere

Reducing A PlaneTo The EquivalentHorizontal

Incline the plane by 80d, dragging the spike within the sphere’s surface.

i

Reducing A PlaneTo The EquivalentHorizontal

Decline the plane by 40 d. This is a rotation about the vertical at your site.

i

d

d

Reducing A PlaneTo The EquivalentHorizontal

i

d

d

h

C

Lat. 42

Inc. 80

Dec. 40

Eq. Lat. -26.4

Inc. Merid. 44.9

Slope -32.2

Z

P S

Thank you !