Spherical Trigonometery

I am trying to apply the formula

cos c = cos a x cos b + sin a x sin b x cos C

to find the length of c in my spherical triangle.

I am working with 2 examples in a book in which the answers are given.

In the first example all the sines & cosines calculated are positive and I agree with the answer in the book no problem.

In the second example one of the angles is 100 degrees and gives a negative cosine (cosine C). In this case I do not agree with the solution in the book but believe it is that I am unaware what to do with the negative cosine.

Not sure if this will help but triangle CBA

Angle C is known 100 degrees

Length of a is known 34 degrees

Length of b is known 38 degrees

As the sphere is the earth 1 degree = 60 nautical mile.

The answer given for length c is 54 degrees which multiplied by 60 = 3240 nautical miles.

I am unable to discover where 54 degrees comes from but feel it is the negative cosine confusing me.

Please help



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  1. Navigator reporting. Dory is bailed out and oars shipped.
    I agree with the 54 degrees.
    Wicked ice on the harbor this morning. Had to break crust to get to breakfast up at Cripple Cove landing.
    Sorry about being late for watch :)
    cos c = cos a cos b + sin b sin a cos C

    cos c = .829*.788 + .61566*.559 cos C

    cos c = .653 + .344 ( -.1736)

    cos c = .653 - .0597

    cos c = .593

    c = 53.6 degrees

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  2. You probably were messed up by the angle(distance) c being wanted rather than a as the law is usually written.
    Best to remember it using this picture in your head:

    cos side unknown = cos other side*cos third side + sin other side*sin third side* cos angle opposite unknown side

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  3. Thanks again Damon.

    Problem is I cannot get my calculator to come up with -.0597 by using .344(-.1736)?

    All my figures for sines and cosines are the same as yours but I obviously do not know how to use my calculator!

    .344 x -.1736 gives me .1704 on my calculator?


    -.1736 x .344 = .0597 eurika!

    Is it always necessary to enter the negative values first? or is it my calculator?

    Thanks again


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  4. Your calculator probably does not like getting two orders at once, like negative and multiply. It was the x- that confused it.

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