Question 205115
just about everything you would ever want to know about the law of cosines alnd probably lots more can be found at this web address:
http://en.wikipedia.org/wiki/Law_of_cosines
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the law of cosines is a generalization of the pythagorean theorem that applies to all triangles regardless if they are right triangles or not.  The pythagorean formula applies to right triangles only.
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the pythagorean theorem states:
{{{c^2 = a^2 + b^2}}}
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the law of cosines theorem states:
{{{c^2 = a^2 + b^2 - 2*a*b*cos(c)}}}
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your problem states:
Two cyclists leave the corner of State Street and Main Street simultaneously.  State Street and Main Street are not at right angles;  the cyclists' paths have an angle of  45o  between them.  How far apart are the cyclists after they each travel 7 miles?  Hint:  Use the Law of Cosines
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Your triangle is ABC
side opposite angle A is side a.
side opposite angle B is side b.
side opposite angle C is side c.
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since side a and side b are equal, they form an isosceles triangle allowing you to solve this without use of the law of cosines.
we will, however, solve it first using the law of cosines.
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we have:
side a = 7
side b = 7
angle c = 45 degrees.
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law of cosines formula is:
{{{c^2 = a^2 + b^2 - 2*a*b*cos(c)}}}
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substituting known values in that formula we get:
{{{c^2 = 7^2 + 7^2 + 2*7*7*cos(45)}}}
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cos(45) = .707106781...
formula becomes:
{{{c^2 = 7^2 + 7^2 + 2*7*7*(.707106781)}}}
solving we get:
{{{c^2 = 49 + 49 + 98*(.707106781)}}}
which becomes:
{{{c^2 = 28.70353544}}}
take square root of both sides to get:
{{{c = 5.357568053}}}
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distance between the cyclists is 5.36 miles.
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a picture can be found at the following website:
http://theo.x10hosting.com/