Cosine theorem
We want to find the side AC.
We can use the cosine theorem if we have the lengths of the other two sides (AB and BC) and the value of its opposite angle (∡B):
STEP 1: theorem equation
We have that the formula of the Cosine theorem is:
[tex]AC^2=AB^2+BC^2-2(AB)(BC)\cos (∡B)[/tex]
STEP 2: replacing in the equation
Since,
AB = 23
BC = 8
and
∡B = 110º
We can find AC by replacing those values in the formula:
[tex]\begin{gathered} AC^2=AB^2+BC^2-2(AB)(BC)\cos (∡B) \\ \downarrow \\ AC^2=23^2+8^2-2(23)(8)\cos (110º) \end{gathered}[/tex]
STEP 3: solving the operation
Since
cos(110º) = -0.342
and
23² = 529
8² = 64
then
[tex]\begin{gathered} AC^2=23^2+8^2-2(23)(8)\cos (110º) \\ \downarrow \\ AC^2=529+64-2(23)(8)(-0.342) \end{gathered}[/tex]
Since
2(23)(8)(-0.342)= -125.856
[tex]\begin{gathered} AC^2=529+64-2\mleft(23\mright)\mleft(8\mright)\mleft(-0.342\mright) \\ \downarrow \\ AC^2=529+64-(-125.856) \\ =529+64+125.856 \\ =718.856 \end{gathered}[/tex]
Squaring root both sides, we have:
[tex]\begin{gathered} AC^2=718.856 \\ \downarrow \\ \sqrt{AC^2}=AC=\sqrt{718.856} \\ =26.8115 \end{gathered}[/tex]
Then, AC length is approximately 26.8116 in.
ANSWER: D
