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To solve the right triangle \(ABC\), where \( C = 90^\circ \), with the given values \(a = 19.3 \, \text{cm}\) and \(c = 46.2 \, \text{cm}\), we will use trigonometric relationships and the Pythagorean theorem.

### Step 1: Find \(b\)
Using the Pythagorean theorem:
\[
b^2 = c^2 - a^2
\]
\[
b = \sqrt{c^2 - a^2} = \sqrt{46.2^2 - 19.3^2}
\]

### Step 2: Find angle \(A\)
To find angle \(A\), we use the sine function:
\[
\sin(A) = \frac{a}{c} = \frac{19.3}{46.2}
\]
Now calculate \(A\) by finding the inverse sine.

### Step 3: Find angle \(B\)
Since the sum of angles in a triangle is \(180^\circ\), and \(C = 90^\circ\), we can calculate \(B\) as:
\[
B = 90^\circ - A
\]

I'll calculate these values for you next.Here are the results for the right triangle:

- \( b \approx 42.0 \, \text{cm} \) (rounded to the nearest tenth)
- \( A = 24^\circ 41' \) (rounded to the nearest minute)
- \( B = 65^\circ 18' \) (rounded to the nearest minute)

Would you like more details on any of the steps, or do you have any further questions?

Here are five related questions you can explore:
1. How do you solve a right triangle when given two angles?
2. How do you use trigonometric functions to find missing sides of a triangle?
3. What is the difference between sine, cosine, and tangent?
4. How can the Pythagorean theorem be applied in different contexts?
5. How do you convert decimal degrees into degrees and minutes?

**Tip:** When working with trigonometric functions, always ensure your calculator is set to the correct mode (degrees or radians) depending on the problem.

Solve the right triangle ABC, where C = 90°, with a = 19.3 cm and c = 46.2 cm. Find b, A, and B (rounded to the nearest tenth and minute).

Learn how to solve a right triangle with sides a = 19.3 cm, c = 46.2 cm. Use the Pythagorean theorem to find the missing side and trigonometric functions to find angles A and B, rounded to the nearest minute.

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