solve right triangle abc with c=90 degrees, a = 31.1 degrees and b = 33.1cm

We are tasked with solving a right triangle \( \triangle ABC \), where:

- \( C = 90^\circ \) (right angle),
- \( A = 31.1^\circ \),
- \( b = 33.1 \, \text{cm} \) (the side opposite angle \( B \)).

To solve the triangle, we need to find:

1. Side \( a \) (opposite angle \( A \)),
2. Side \( c \) (the hypotenuse),
3. Angle \( B \) (since \( B = 90^\circ - A \), this is straightforward).

### Step 1: Find angle \( B \)

Since the sum of angles in a triangle is \( 180^\circ \), we can calculate:

\[
B = 90^\circ - A = 90^\circ - 31.1^\circ = 58.9^\circ
\]

### Step 2: Use trigonometric ratios to find side \( c \)

We can use the sine function to find the hypotenuse \( c \). The sine of angle \( B \) relates the opposite side \( b \) and the hypotenuse \( c \):

\[
\sin(B) = \frac{b}{c}
\]

Substituting the known values:

\[
\sin(58.9^\circ) = \frac{33.1}{c}
\]

\[
c = \frac{33.1}{\sin(58.9^\circ)}
\]

Let me calculate this.

The hypotenuse \( c \) is approximately \( 38.66 \, \text{cm} \).

### Step 3: Use trigonometric ratios to find side \( a \)

Now we can use the sine function for angle \( A \) to find side \( a \), which is opposite angle \( A \):

\[
\sin(A) = \frac{a}{c}
\]

Substituting the known values:

\[
\sin(31.1^\circ) = \frac{a}{38.66}
\]

\[
a = 38.66 \times \sin(31.1^\circ)
\]

Let me calculate \( a \).

# Given angle A
A = 31.1  # angle A in degrees

# Calculate side a using the sine function
a = c * math.sin(math.radians(A))
aSide \( a \) is approximately \( 19.97 \, \text{cm} \).

### Summary of the solution:

- \( A = 31.1^\circ \)
- \( B = 58.9^\circ \)
- \( C = 90^\circ \)
- \( a \approx 19.97 \, \text{cm} \)
- \( b = 33.1 \, \text{cm} \)
- \( c \approx 38.66 \, \text{cm} \)

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How do you use the Pythagorean theorem to verify the solution?
2. What are other trigonometric identities that can be used to solve triangles?
3. How does the Law of Sines apply to non-right triangles?
4. How can you find the area of a right triangle using the sides?
5. How do you solve a triangle if only two sides are given?

**Tip:** In a right triangle, the sum of the non-right angles always equals \( 90^\circ \).

Learn how to solve a right triangle ABC where C is 90°, A is 31.1°, and side b = 33.1 cm. This problem involves calculating angles and sides using trigonometric ratios like sine, cosine, and solving for the hypotenuse. Suitable for high school students in Grades 9-12.

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