I have three points with coordinates: $A(5,-1,0),B(2,4,10)$ and $C(6,-1,4)$

I have the following vectors $\overrightarrow{CA}=(-1,0,-4)$ and $\overrightarrow{CB}=(-4,5,6)$

To find the area of the triangle I used the dot product between these vectors to get the angle and then applied the formula $A=0.5ab\mathrm{sin}C$ to find the area of the triangle which gave me 15.07(2dp).

However in the given solutions the answer is given as $(3\ast \sqrt{(}102))/2$

I think they have used the trig identity ${\mathrm{cos}}^{2}(\theta )+{\mathrm{sin}}^{2}(\theta )=1$ to find the value of $\mathrm{sin}(\theta )$ rather than $\mathrm{arccos}(\theta )$ to find the angle ACB. However I don't understand why there would be such a discrepancy between the two answers; one using $\mathrm{arccos}$ and the other using the trig identity.

I have the following vectors $\overrightarrow{CA}=(-1,0,-4)$ and $\overrightarrow{CB}=(-4,5,6)$

To find the area of the triangle I used the dot product between these vectors to get the angle and then applied the formula $A=0.5ab\mathrm{sin}C$ to find the area of the triangle which gave me 15.07(2dp).

However in the given solutions the answer is given as $(3\ast \sqrt{(}102))/2$

I think they have used the trig identity ${\mathrm{cos}}^{2}(\theta )+{\mathrm{sin}}^{2}(\theta )=1$ to find the value of $\mathrm{sin}(\theta )$ rather than $\mathrm{arccos}(\theta )$ to find the angle ACB. However I don't understand why there would be such a discrepancy between the two answers; one using $\mathrm{arccos}$ and the other using the trig identity.