SOLUTION: 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 bet

Question 205115: 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
Can you explain how the law of cosines works and how it would apply to this problem?
Thank You
Found 2 solutions by Alan3354, Theo:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
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
Can you explain how the law of cosines works and how it would apply to this problem?
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http://hyperphysics.phy-astr.gsu.edu/Hbase/lcos.html
c^2 = 7^2 + 7^2 - 2*7*7*cos(45)
c^2 = 98 - 98sqrt(2)/2
c^2 = 98-49sqrt(2)
c^2 =~ 98 - 69.296
c^2 =~ 28.7
c =~ 5.358
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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:

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the law of cosines theorem states:

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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:

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substituting known values in that formula we get:

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cos(45) = .707106781...
formula becomes:

solving we get:

which becomes:

take square root of both sides to get:

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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/
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