You have been asked to set up a satellite receiver in a remote location. In your

Question

You have been asked to set up a satellite receiver in a remote location. In your office, you determined the appropriate coordinate direction angles to orient the antenna as needed. These were alpha = 81.2degree, beta = 35.4 degree, and gamma = 56.0 degree. However, when you got into the field, you found out the instrumentation for the antenna requires the angles theta_h and theta_v as shown in Figure 1 below. You are in a remote location...all you have is a pencil, pad of paper, and your calculator. What are the appropriate values for theta_h and theta_v? middot Ignore the "Height off ground" (see Figure 1). In other words, don't use the coordinate system given in Figure 1. It's easier to work this problem if you place the origin of your system at point 0. middot Model the antenna as a unit vector. That way you don't need to know its length. Express the unit vector using the coordinate direction angles provided (see Figure 2). Note that the figures are not to scale. middot Then use geometry/trig to compute theta_h and theta_v (see Figure 3).

Explanation / Answer

Since we have taken unit vector, we can need not to write length

theta v = pi/2 - gamma

cos beta = [cos thetav ] cos theta h

cos theta h = cos beta / [cos thetav ]

                    = cos beta / [cos(pi/2 - gamma) ]

                    = cos beta/sin gamma

                    = cos beta cosec gamma

theta h = cos -1 [ cos beta cosec gamma ]

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