1. A C B
Introduction To Spherical Trigonometry/Astronomy
Fred Sawyer Chicago 2005

2. A C B

3. b A C c a B
Rule 1.
The sum of the lengths of a spherical triangle's sides is
always less than 360º.

4. b A C c a B
Rule 2.
The sum of the angles at its vertices is greater than 180º
and less than 540º.

5. b A C c a B
Rule 3.
The sum of the lengths of any two sides is greater
than the length of the third side.

6. b A C c a B
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.

7. b A C c a B
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.

8. b A C
The Fundamental Law
Of Spherical Trigonometry c a B
Cosine Law

9. b A C
The Fundamental Law
Of Spherical Trigonometry c a B
Cosine Law

10. b A C
The Fundamental Law
Of Spherical Trigonometry c a B
Cosine Law

11. b A C c a B
Sine Law

12. b A C c a B
Four Element Equation

13. b A C c a B
Given any 3 elements of a spherical triangle,
it is possible to solve for the other 3.

14. b A C c a B

15. b C a B
Four Element Equation

16. b A
Most Difficult Case a B
But… Is B acute or obtuse???
Appeal to Rules 4 and 5

17. b A c a B

18. b A c a B

19. P S Z
Given latitude, declination, and hour angle –
Find solar altitude.

20. P S Z

21. P S Z
Given altitude, azimuth and latitude,
Find the solar declination and hour angle.

22. P S Z
Cosine law gives us declination from altitude, azimuth and latitude.
Then sine law gives us hour angle from declination, azimuth and altitude.

23. P S Z
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.

24. P S Z
Given latitude, hour angle, and declination,
Find the sun’s azimuth.

25. P S Z

26. P S Z

27. H P Z S

28. P S Z

29. P S Z
Prosthaphaeretical Arc
Hectemoral Arc (complement)

30. Reducing A Plane
To The Equivalent
Horizontal
Begin with a horizontal plane at latitude
Spike the celestial sphere

31. Reducing A Plane
To The Equivalent
Horizontal
Incline the plane by 80d, dragging the spike within the sphere’s surface.

32. Reducing A Plane
To The Equivalent
Horizontal
Decline the plane by 40 d. This is a rotation about the vertical at your site.

33. Reducing A Plane
To The Equivalent
Horizontal
Lat. 42
Inc. 80
Dec. 40
Eq. Lat
Inc. Merid. 44.9
Slope

34. P S Z
Thank you !