Based on the uploaded image, let’s break down the problem step by step.

Question 1

For the triangle with:

• $m\angle C = 13^\circ$,
• $m\angle A = 22^\circ$,
• $c = 9$, find side $a$ to the nearest whole number.

Solution:

1. Use the Triangle Sum Rule:
   $m\angle A + m\angle B + m\angle C = 180^\circ$
   Substitute the values:
   $22^\circ + m\angle B + 13^\circ = 180^\circ$
   Solve for $m\angle B$:
   $m\angle B = 180^\circ - 22^\circ - 13^\circ = 145^\circ$

2. Apply the Law of Sines: $\frac{a}{\sin A} = \frac{c}{\sin C}$ Rearrange to solve for $a$: $a = c \cdot \frac{\sin A}{\sin C}$

   Substitute the given values:
   $c = 9$, $\sin A = \sin(22^\circ)$, $\sin C = \sin(13^\circ)$.

3. Use a calculator to find the values:
   $\sin(22^\circ) \approx 0.3746, \quad \sin(13^\circ) \approx 0.2249$
   Substitute into the equation: $a = 9 \cdot \frac{0.3746}{0.2249} \approx 9 \cdot 1.666 \approx 15$

Answer: $a = 15$

Question 2

For the triangle where:

• $m\angle C = 145^\circ$,
• $b = 7$,
• $c = 33$, find $m\angle B$ rounded to the nearest degree.

Solution:

1. Apply the Law of Cosines: $\cos B = \frac{b^2 + c^2 - a^2}{2bc}$ Rearrange to solve for $\cos B$:
   Since $a$ is not provided, we need more clarification on this part or use approximations.

Would you like help refining the specific process or additional clarification for this problem?

Follow-Up Questions:

1. Would you like more detailed steps on the calculations for the second problem?
2. Do you need an explanation of the Law of Sines or Law of Cosines for review?
3. Shall we verify the solution for the first problem with additional context?
4. Would you like help setting up similar problems for practice?
5. Is there a specific trigonometric question you'd like to focus on?

Tip:

Always label your angles and sides clearly when applying the Laws of Sines or Cosines to avoid mixing up values!