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use SIne or cosine rule to solve and show the steps for both of the question

Let's solve both questions using the appropriate trigonometric rules.

### Problem 17: Finding Side \( BC \) Using the Law of Cosines

Given:
- \( AB = 20 \, \text{in} \)
- \( AD = 36 \, \text{in} \)
- \( \angle DAB = 30^\circ \)

To find \( BC \) in the parallelogram:

In a parallelogram, opposite sides are equal. Therefore, \( BC = AD = 36 \, \text{in} \).

However, for accuracy, we use the Law of Cosines:

\[
BC^2 = AB^2 + AD^2 - 2 \times AB \times AD \times \cos(\angle DAB)
\]

Substituting the given values:

\[
BC^2 = 20^2 + 36^2 - 2 \times 20 \times 36 \times \cos(30^\circ)
\]

\[
BC^2 = 400 + 1296 - 1440 \times \cos(30^\circ)
\]

Since \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \):

\[
BC^2 = 1696 - 1440 \times 0.866
\]

\[
BC^2 = 1696 - 1247.04
\]

\[
BC^2 = 448.96
\]

\[
BC \approx \sqrt{448.96} \approx 21.19 \, \text{in}
\]

So, \( BC \) to the nearest tenth is **21.2 in**.

### Problem 18: Finding Angle \( \angle DBC \) Using the Law of Sines

Given:
- \( AB = 25 \, \text{cm} \)
- \( AD = 16 \, \text{cm} \)
- \( \angle DAB = 120^\circ \)

To find \( \angle DBC \):

First, we can use the Law of Cosines to find \( BD \):

\[
BD^2 = AB^2 + AD^2 - 2 \times AB \times AD \times \cos(120^\circ)
\]

Substitute the given values:

\[
BD^2 = 25^2 + 16^2 - 2 \times 25 \times 16 \times \cos(120^\circ)
\]

Since \( \cos(120^\circ) = -\frac{1}{2} \):

\[
BD^2 = 625 + 256 + 400
\]

\[
BD^2 = 1281
\]

\[
BD \approx \sqrt{1281} \approx 35.79 \, \text{cm}
\]

Now using the Law of Sines to find \( \angle DBC \):

\[
\frac{\sin(\angle DBC)}{AD} = \frac{\sin(\angle DAB)}{BD}
\]

Substitute the values:

\[
\frac{\sin(\angle DBC)}{16} = \frac{\sin(120^\circ)}{35.79}
\]

\[
\sin(\angle DBC) \approx 16 \times \frac{0.866}{35.79}
\]

\[
\sin(\angle DBC) \approx 0.387
\]

\[
\angle DBC \approx \arcsin(0.387) \approx 22.8^\circ
\]

Rounding to the nearest whole degree, \( \angle DBC \) is **23°**.

---

### Do you want more details or have any questions? Here are five related questions to further your understanding:

1. How does the Law of Cosines help in solving for unknown sides in a triangle?
2. Why is the cosine of \( 120^\circ \) negative?
3. Can the Law of Sines be used if only one angle is given in a triangle?
4. What is the significance of \( \sqrt{3}/2 \) in trigonometric calculations?
5. How does the area of a parallelogram relate to its side lengths and angles?

**Tip:** When dealing with trigonometry in non-right triangles, always check if the Law of Sines or Cosines is more appropriate for the given data.

Learn how to solve trigonometry problems using the Law of Cosines and Law of Sines with detailed step-by-step solutions. These problems involve finding sides and angles in non-right triangles. Suitable for high school students.

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