# A triangle has sides A, B, and C. Sides A and B are of lengths 5 and 6, respectively, and the angle between A and B is (5pi)/12 . What is the length of side C?

Jan 5, 2016

c = 6.74 ( 2 decimal places )

#### Explanation:

I recommend that you draw a sketch of the triangle where you will see that we are given 2 sides and the included angle ie. the angle between the 2 given sides.

For this condition to find the third side use the cosine rule

 color(red)( c^2 = a^2 + b^2 - ( 2ab cos c )

here $a = 5 , b = 6 \mathmr{and} c = \frac{5 \pi}{12}$

substitute these values into the cosine rule

 rArr c^2 = 5^2 + 6^2 - ( 2 xx 5 xx 6 xx cos((5pi)/12 )

evaluating : ${c}^{2} = 25 + 36 - 15.529$

$\Rightarrow {c}^{2} = 45.471$

At this stage remember to take the square root to get c.

$\Rightarrow c = 6.74$ ( 2 decimal places )