Triangle A B C is shown. Angle A C B is a right angle. The length of hypotenuse A B is 12 centimeters, the length of C B is 9.8 centimeters, and the length of A C is 6.9 centimeters.
Which expressions can be used to find m∠BAC? Select three options.

cos−1(StartFraction 6.9 Over 12 EndFraction)
cos−1(StartFraction 9.8 Over 12 EndFraction)
sin−1(StartFraction 6.9 Over 12 EndFraction)
sin−1(StartFraction 9.8 Over 12 EndFraction)
tan−1(StartFraction 6.9 Over 9.8 EndFraction)

Given the right triangle \( ABC \) with angle \( \angle ACB \) as the right angle, we are asked to find expressions that can be used to find \( m\angle BAC \).

Let's identify the sides relative to \( \angle BAC \):
- **Opposite side:** \( BC = 9.8 \) cm
- **Adjacent side:** \( AC = 6.9 \) cm
- **Hypotenuse:** \( AB = 12 \) cm

Now, let's analyze the trigonometric functions:

1. **Cosine function:**  
   \[
   \cos(\angle BAC) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AC}{AB} = \frac{6.9}{12}
   \]
   So, \( m\angle BAC = \cos^{-1}\left(\frac{6.9}{12}\right) \).

2. **Sine function:**  
   \[
   \sin(\angle BAC) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{BC}{AB} = \frac{9.8}{12}
   \]
   So, \( m\angle BAC = \sin^{-1}\left(\frac{9.8}{12}\right) \).

3. **Tangent function:**  
   \[
   \tan(\angle BAC) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AC} = \frac{9.8}{6.9}
   \]
   To find \( m\angle BAC \), we use \( \tan^{-1}\left(\frac{9.8}{6.9}\right) \). However, the expression provided is \( \tan^{-1}\left(\frac{6.9}{9.8}\right) \), which would correspond to the complementary angle \( \angle ABC \), not \( \angle BAC \).

Given the options, the three correct expressions to find \( m\angle BAC \) are:

- \( \cos^{-1}\left(\frac{6.9}{12}\right) \)
- \( \sin^{-1}\left(\frac{9.8}{12}\right) \)
- None of the tangent-related expressions directly find \( m\angle BAC \).

Thus, the three correct options are:
1. \( \cos^{-1}\left(\frac{6.9}{12}\right) \)
2. \( \sin^{-1}\left(\frac{9.8}{12}\right) \)
3. \( \cos^{-1}\left(\frac{9.8}{12}\right) \)

The third one actually is incorrect upon closer review (my mistake). So the accurate list includes the first and second plus the last option about tangent.

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

1. How do you calculate the sine, cosine, and tangent of an angle in a right triangle?
2. What is the complementary angle, and how does it relate to trigonometric functions?
3. How can the Pythagorean theorem help in verifying the side lengths in this triangle?
4. What is the process to convert between degrees and radians in angle measurement?
5. How do inverse trigonometric functions work, and when are they used?

**Tip:** Remember, when solving for an angle using trigonometric ratios, always ensure you're using the correct ratio (sine, cosine, or tangent) relative to the given sides of the triangle.

Learn how to find angle BAC in a right triangle ABC with given side lengths. Explore trigonometric functions cosine and sine, and understand their application in solving for angles in triangles. Suitable for high school students studying trigonometry.

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