Angles of a triangle in radians

Question

For some reason I can't figure out two of these angles from a triangle. I was able to determine the length of the triangles sides and with that I could figure out θb= 1.107 I thought with that I would easily be able to determine the other two angles but every possible answer I calculated is wrong. Can somebody please help me out?

A triangle is defined by the three points:

A=(6,9)

B=(9,8)

C=(5,4)

Determine all three angles in the triangle (in radians).

θa=__________ θb=1.107 θc=__________

Answer 1

Using https://en.wikipedia.org/wiki/Law_of_cosines, and calling writing $a = BC$, $b= AC$ and $c=AB$ we find (hopefully I didn't make any errors in my calculations, which you should check):

$73 = c^2 = 20 + 32 - 2\sqrt{20}\sqrt{32} \cos \gamma$

from which we find that $\cos \gamma = \frac{21}{-2\sqrt{20}\sqrt{32}}$ and using a calculator, we find that $\gamma = 1.8492$ radians. (Or $105.9$ degrees)

The other angles can be calculated similarly.

Answer 2

Hint use

$$\frac{AB}{\sin(\theta_c)}=\frac{BC}{\sin(\theta_a)}=\frac{CA}{\sin(\theta_b)}$$