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).