To find the expressions that can be used to determine $m\angle ABC$ in the right triangle $ABC$, let's identify the sides relative to $\angle ABC$:

• Opposite side to $\angle ABC$: $AC = 6.9$ cm
• Adjacent side to $\angle ABC$: $BC = 9.8$ cm
• Hypotenuse: $AB = 12$ cm

Now, let's consider the possible trigonometric functions:

1. Cosine function: $\cos(\angle ABC) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AB} = \frac{9.8}{12}$ So, $m\angle ABC = \cos^{-1}\left(\frac{9.8}{12}\right)$.

2. Sine function: $\sin(\angle ABC) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AC}{AB} = \frac{6.9}{12}$ So, $m\angle ABC = \sin^{-1}\left(\frac{6.9}{12}\right)$.

Given this information, the two correct expressions that can be used to find $m\angle ABC$ are:

1. $\cos^{-1}\left(\frac{9.8}{12}\right)$
2. $\sin^{-1}\left(\frac{6.9}{12}\right)$

These correspond to the cosine of the angle based on the adjacent and hypotenuse and the sine of the angle based on the opposite and hypotenuse.

If you want more details or have any questions, feel free to ask! Here are some related questions:

1. How can you determine which trigonometric function to use based on given sides?
2. What is the relationship between sine and cosine for complementary angles?
3. How do inverse trigonometric functions help in finding angle measures?
4. What steps would you follow to solve for an unknown angle in any right triangle?
5. How does the Pythagorean identity relate to sine and cosine functions?

Tip: Always remember that the adjacent side for any angle in a right triangle is the side next to the angle that is not the hypotenuse, while the opposite side is the one directly across from the angle.