Question

in the figure AB equals how many inches AC equals how many inches

Answer 1

Remark 
This is a trig question. You need two function (Tan(C) and Cos(C)

Tan(C)
this will give you the value for BA
Tan(C) = BA / BC

Givens
C = 40 degrees
BC = 10
tan(C) = AB/BC
Tan(40) = AB / 10
0.8391 = AB / 10 Multiply both sides by 10
10*0.8391 = AB
AB = 8.391 

Cos(C)
Cos(C) = BC/AC
BC = 10
C = 40 degrees.
Cos(C) = 0.76604
0.76604 = 10/AC     Multiply both sides by AC
0.76604 * AC = 10
AC = 10/0.76604
AC = 13.054

Answer 2

Answer: AB = 8.4 inches   and   AC= 13 inches

Step-by-step explanation:

In the given picture, we have a right triangle.

The sides adjacent to $40^{\circ}=10\ inches$

Applying trigonometry, we have

$\cos 40^{\circ}=\dfrac{\text{sides adjacent to } 40^{\circ}}{\text{Hypotenuse}}\\\\\Rightarrow\ 0.76604444311=\dfrac{10}{H}\\\\\Rightarrow\ H=13.0540728935\approx13\  inches$

Thus, AC= 13 inches

Also,

$\tan 40^{\circ}=\dfrac{AB}{10}\\\\\Rightarrow AB=10\times0.83909963117\\\\\Rightarrow\ AB=8.3909963117\approx8.4\ inches$