Figure: Determining distances with trigonometry

Caption: An image of a right triangle illustrating the meaning of some common terms of trigonometry: adjacent, opposite, hypotenuse. The angles of the right triangle are A = 30°, B = 60°, and C = 90°.

We use the image to explicate how to determine distances using trigonometry.

Further explication:

2. Angle θ (A in the image) is a parallax if it were measured by moving along the opposite (which is the baseline for the parallax) and measuring the shift in angular position of vertex A (which is just angle A itself) against the background of remote reference points.

   We can calculate distance b using angle θ and baseline distance "a" using trigonometry:

     tan(θ)=a/b , therefore  b=a/tan(θ)  .  

   If angle A is very small, we can make the small-angle approximation tan(θ)≅θ and obtain to 1st-order in small θ

     b=a/θ  , 

   where angle θ is measured in radians.

   Radians are the natural units for measuring angles---but they are NOT always convenient units since there are an irrational number of them in a circle: i.e., 2π = 6.283185307179586 .... The conversion formula from degrees to radians is

     θ(radians) = θ(°)*(2π radians/360°).  

   Astro-bodies are usually so remote that their parallaxes are tiny for any baseline that we can use. Thus, the small-angle approximation virtually always gives negligible error in astronomy.