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

Step-by-step explanation:

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

+) cosine of an angle = $\frac{Adjacent side 1^{2}+Adjacent side 2^{2} - Opposite side^{2}}{2*Adjacent side 1 * Adjacent side 2}$

Angel B = 30° has:

+) 2 adjacent sides: BC and AB

+) Opposite side: AC

=> We have:

cosine ∠B = $\frac{BC^{2}+AB^{2}-AC^{2} }{2*AB*BC}$

=> cosine 30°  = $\frac{(2\sqrt{3})^{2} + 4^{2} - AC^{2} }{ 2 *4*2\sqrt{3} }$

=> $\frac{\sqrt{3} }{2} = \frac{28 - AC^{2} }{16\sqrt{3} }$

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

=> AC^2 = 4

We have:

$BC^{2} = (2\sqrt{3} )^{2} =12$

$AB^{2} = (4 )^{2} =16$

$AC^{2} = 4$

=> $AC^{2} +BC^{2} = AB^{2}$

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

=> X + 30 = 90

=> X = 60°