To find side $b$ of the triangle, given:

• $m\angle C = 13^\circ$,
• $m\angle A = 22^\circ$,
• $c = 9$ (side opposite $\angle C$),

we can use the Law of Sines. This law states that in any triangle:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Step 1: Find $\angle B$

Since the sum of angles in a triangle is $180^\circ$, we can calculate $m\angle B$ as follows:

$m\angle B = 180^\circ - m\angle A - m\angle C$ $m\angle B = 180^\circ - 22^\circ - 13^\circ = 145^\circ$

Step 2: Apply the Law of Sines to Find $b$

Now, using the Law of Sines:

$\frac{b}{\sin B} = \frac{c}{\sin C}$

Rearranging to solve for $b$:

$b = c \cdot \frac{\sin B}{\sin C}$

Substitute the known values:

$b = 9 \cdot \frac{\sin 145^\circ}{\sin 13^\circ}$

Step 3: Calculate $b$ (using approximate values)

1. $\sin 145^\circ \approx 0.5736$
2. $\sin 13^\circ \approx 0.2249$

$b \approx 9 \cdot \frac{0.5736}{0.2249} \approx 9 \cdot 2.55 \approx 22.95$

Answer

Rounding $b$ to the nearest whole number:

$b \approx 23$

Would you like further explanation on any part of this solution?

Here are some related questions to explore:

1. How would you find side $a$ in this triangle?
2. What if the triangle were not solvable using the Law of Sines?
3. How does the Law of Sines differ from the Law of Cosines?
4. What other methods can be used to solve for sides in a triangle?
5. Why is it important to check for ambiguous cases in triangle problems?

Tip: When using trigonometric laws, remember that calculator settings (degrees vs. radians) affect your results.