Question

I really need a lot of help with this question. Answer step by step and well and explain every conclusion you've got.

Obs : The answer is 60°, but, i need to understand, if possible, why this is the value

Answer 1

Explanation:

In a normal triangle, there is a formula as following:

+) cosine of an angle =
`(Adjacent side 1^(2)+Adjacent side 2^(2) - Opposite side^(2))/(2*Adjacent side 1 * Adjacent side 2)`

Angel B = 30° has:

+) 2 adjacent sides: BC and AB

+) Opposite side: AC

=> We have:

cosine ∠B =
`(BC^(2)+AB^(2)-AC^(2) )/(2*AB*BC)`

=> cosine 30° =
`((2√(3))^(2) + 4^(2) - AC^(2) )/( 2 *4*2√(3) )`

=>
`(√(3) )/(2) = (28 - AC^(2) )/(16√(3) )`

=> 2 x (28 - AC^2) = 48

=> AC^2 = 4

We have:

`BC^(2) = (2√(3) )^(2) =12`

`AB^(2) = (4 )^(2) =16`

`AC^(2) = 4`

=>
`AC^(2) +BC^(2) = AB^(2)`

According the inverse Pythagorean theorem, ABC is the right angle triangle with Angle ACB = 90°

=> X + 30 = 90

=> X = 60°