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Round your answer to the nearest hundredth 

To determine the measure of angle \( \angle B \) in the given right triangle \( \triangle ABC \) where \( AB \) is the hypotenuse, \( AC = 7 \) is one leg, and \( BC = 5 \) is the other leg, we will use trigonometric ratios.

Given:
- \( AC = 7 \)
- \( BC = 5 \)

We can use the tangent function, which relates the opposite side to the adjacent side of a right triangle:
\[ \tan(\angle B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AC}{BC} \]

Thus,
\[ \tan(\angle B) = \frac{7}{5} \]

To find \( \angle B \), we take the arctangent (inverse tangent) of \( \frac{7}{5} \):
\[ \angle B = \tan^{-1}\left(\frac{7}{5}\right) \]

Using a calculator,
\[ \angle B \approx \tan^{-1}(1.4) \approx 54.46^\circ \]

Rounded to the nearest hundredth, the measure of \( \angle B \) is:
\[ \angle B \approx 54.46^\circ \]

Would you like more details or have any questions about this solution?

Here are 8 related questions to further your understanding:

1. How do you use the sine function to find \( \angle B \)?
2. What is the measure of \( \angle C \) in the triangle?
3. Can you find the length of the hypotenuse \( AB \) using the Pythagorean theorem?
4. How do you convert \( 54.46^\circ \) to radians?
5. What is the cosine of \( \angle B \)?
6. If the triangle was scaled up by a factor of 2, what would the new side lengths be?
7. How do you verify the result using a trigonometric identity?
8. What are the properties of a right triangle?

**Tip:** Always double-check your calculations and make sure your calculator is set to the correct mode (degrees or radians) based on the problem requirements.

Learn how to find the measure of angle B in a right triangle ABC where AC = 7, BC = 5 using trigonometric ratios and the arctangent function. Understand the application of tangent and arctangent functions in trigonometry. Suitable for students in Grades 10-12.

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