https://file.aiquickdraw.com/mathbot/file/1727274288174apbiv4oa.jpg

Let's solve the right triangle based on the given information.

We are told that \( \sin B = \frac{6}{9} \), which simplifies to \( \sin B = \frac{2}{3} \).

### Step 1: Find Angle \( B \)
To find the measure of angle \( B \), we can use the inverse sine function:
\[
B = \sin^{-1}\left(\frac{2}{3}\right)
\]
Let me calculate this:

The measure of angle \( B \) is approximately \( 42^\circ \) (rounded to the nearest whole number).

### Step 2: Find Angle \( A \)
Since the triangle is a right triangle, the sum of the angles must be \( 180^\circ \). Thus:
\[
A = 90^\circ - B = 90^\circ - 42^\circ = 48^\circ
\]

### Step 3: Find Side Lengths Using Trigonometric Ratios
Let’s now use the trigonometric functions to find the side lengths. We know:

- \( \sin B = \frac{6}{9} = \frac{\text{opposite (BC)}}{\text{hypotenuse (AB)}} \)
- \( \cos B = \frac{\text{adjacent (AC)}}{\text{hypotenuse (AB)}} \)

Given that \( \sin B = \frac{6}{9} \), we can solve for the hypotenuse \( AB \), and then use the Pythagorean theorem to solve for the other sides.

We have:
\[
AB = \frac{9}{\sin B}
\]
Let’s calculate that and then find the remaining sides.

Here are the side lengths of the triangle rounded to the nearest tenth:

- \( AB = 13.5 \) (the hypotenuse)
- \( BC = 9.0 \) (the opposite side to angle \( B \))
- \( AC = 10.1 \) (the adjacent side to angle \( B \))

### Final Answers:
- \( m\angle A = 48^\circ \)
- \( m\angle B = 42^\circ \)
- \( AB = 13.5 \)
- \( BC = 9.0 \)
- \( AC = 10.1 \)

Would you like more details or have any questions?

### Related Questions:
1. How do you apply the cosine rule in non-right triangles?
2. What is the relationship between sine, cosine, and tangent in trigonometry?
3. How can we solve a triangle if only two sides are given?
4. How does the Pythagorean theorem apply in 3D space?
5. What are inverse trigonometric functions used for in real-life applications?

### Tip:
Always double-check the units when calculating trigonometric functions to ensure consistent results (degrees vs radians).

Solve the right triangle with sin(B) = 6/9, side lengths to the nearest tenth, and angles to the nearest whole number.

Solve the right triangle with sin(B) = 6/9, where B is approximately 42 degrees. Find the side lengths and angles using trigonometric functions, including sine and cosine. Calculate AB, BC, and AC to the nearest tenth, and find the angle measures of A and B to the nearest whole number. This problem is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation