Math, Trigonometry

Cate is sailing her boat off the coast, which
runs straight north and south. Her GPS
confirms that she is 8 km from Haytown
and 10 km from Beeville, two towns on the
coast. The towns are separated by an angle
of 80°, as seen from the boat. A helicopter
is hovering at an altitude of 1000 m
halfway between Haytown and Beeville.

b) Determine the angle of elevation of the
helicopter, as seen from the sailboat, to
the nearest tenth of a degree.

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  1. The length between the two towns is 11.67km if that helps

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  2. First we find the distance between (H)aytown and (B)eebille
    AB^2 = 8^2 + 10^2-2(8)(10)cos80°
    AB = 11.67117
    let the midpoint be M
    then BM = 5.835586....

    Let S be the position of the ship,
    again, by the cosine law:
    let's find angle B
    8^2 = 10^2 + 11.67117..^2 - 2(10)(11.67117)cosB
    cosB = .73778..
    B = 42.457°

    Once more by the cosine law in triangle SBM
    SM^2 = 10^2 + 5.835586..^2 - 2(10)(5.835586..)cos42.457
    SM = .... , you do it

    finally we can find our angle
    tan(angle) = 1000/SM = ...
    take tan^-1 to find the angle of elevation

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