A triangle has sides A, B, and C. Sides A and B are of lengths #6# and #3#, respectively, and the angle between A and B is #pi/4#. What is the length of side C?

1 Answer
Shantelle
May 1, 2018

#C = 3sqrt(5-2sqrt2)# or #~~4.42#

Explanation:

When you have the lengths of two sides of a triangle and the angle between, then you can solve the missing side with the law of cosines.

We want side #C#, which can be solved by the Law of Cosines formula #C = sqrt(A^2 + B^2 - 2(A)(B)cosc#, where #cosc# is the angle opposite to side #C#.

Let's substitute in the values and solve:
#C = sqrt((6)^2 + (3)^2 - 2(6)(3)cos(pi/4)#

Now simplify:
#C = sqrt(36 + 9 - 36(sqrt2/2)#

#C = sqrt(45 - 18sqrt2)#

#C = sqrt(9(5-2sqrt2)#

#C = sqrt9 * sqrt(5-2sqrt2)#

#C = 3sqrt(5-2sqrt2)#

You can leave it like that, but if you want the answer to be in decimal form, it is #~~4.42# (rounded to the nearest hundredth's place).

Hope this helps!

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