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How to Solve the PZS Spherical Triangle

The PZS spherical triangle is a special figure found in spherical trigonometry. This triangle is based on the plane of a celestial sphere and has three different vertices: the pole, known as P, the zenith of the observer, also known as Z, and the star that is observed, denoted as S. "Solving" a PZS triangle requires calculating all of the sides and angles of the triangle. In order to do this, you need to know some of the dimensions of the PZS spherical triangle in advance as well as use a calculator to do the trigonometric calculations.

Things You'll Need

Calculator

Instructions

- 1
  Diagram the spherical triangle and note the measurements you already know. In order to solve a PZS spherical triangle, you either need to know the dimensions of the three sides of the triangle or the dimensions of two sides and the angle formed between them. The first technique is known as the altitude method, and the second technique is the hour angle method.
- 2
  Label the remaining parts of the spherical triangle. For the purposes of these calculations, the angles formed at points P, Z and S will be denoted A, B and C. The side that connects A and C is denoted as b, and the side that connects C and B is referred to as a; similarly, B and A are connected by c. The source of the notation for these variables is a 1983 paper published in "Surveying and Mapping"; this paper also includes a diagram that shows a PZS triangle labeled with these values.
- 3
  Use the altitude method. For this technique, you need to know the value of a, b and c. The equation to solve for angle B is cos(B) = ((cos(b) - cos(a) x cos(c))/(sin(a) x sin(c)). Similarly, the equation to solve for angle C is cos(C) = ((cos(c) - cos(a) x cos(b))/(sin(a) x sin(b)). The equivalent calculation for measuring angle A is cos(a) = ((cos(a) - cos(b) x cos(c))/(sin(b) x sin(c)).
- 4
  Use the hour angle technique. For this technique, you need to know the measurements for two sides of the spherical triangle as well as the angle between the two. For the purposes of this step, we will use A, b and c. With these three values you can calculate angle B because Tan(B) = ((sin(A))/(sin(c) x cot (b) - cos(c) x cos(A)). Once you know the values of both A and B, you can calculate the value of C with the following equation: cos (C) = - (cos(A) x cos(B) + sin(A) x sin(B) x cos(c)).

Astronomy