# 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?

May 1, 2018

$C = 3 \sqrt{5 - 2 \sqrt{2}}$ or $\approx 4.42$

#### 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 $\cos c$ 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)

$C = \sqrt{45 - 18 \sqrt{2}}$

C = sqrt(9(5-2sqrt2)

$C = \sqrt{9} \cdot \sqrt{5 - 2 \sqrt{2}}$

$C = 3 \sqrt{5 - 2 \sqrt{2}}$

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

Hope this helps!