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

A C B

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

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

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.

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.

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.

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

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

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

b A C c a B Sine Law

b A C c a B Four Element Equation

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

b A C c a B

b C a B Four Element Equation

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

b A c a B

b A c a B

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

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

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.

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.

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

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H P Z S

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P S Z Prosthaphaeretical Arc Hectemoral Arc (complement)

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

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

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

Reducing A Plane To The Equivalent Horizontal Lat. 42 Inc. 80 Dec. 40 Eq. Lat. -26.4 Inc. Merid. 44.9 Slope -32.2

P S Z Thank you !