https://file.aiquickdraw.com/mathbot/file/17295699296920ozuapks.jpg

The task in the image is to calculate the missing side lengths and angles in triangles based on the given data. For each sub-question, we can apply the **Law of Sines** or **Law of Cosines** depending on the information provided.

Let's break down each part:

### Law of Sines:
\[
\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}
\]
where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(\alpha\), \(\beta\), and \(\gamma\) are the opposite angles.

### Law of Cosines (if needed):
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(\gamma)
\]

### Now, let's go through each sub-part:

---

#### a) Given: \(a = 3.7 \, \text{cm}, c = 6.2 \, \text{cm}, \alpha = 26^\circ\)

To find the missing side \(b\) and angles \(\beta\) and \(\gamma\):
1. Use the Law of Sines to find \(\gamma\):
   \[
   \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)}
   \]
   Rearranging to solve for \(\sin(\gamma)\):
   \[
   \sin(\gamma) = \frac{c \cdot \sin(\alpha)}{a}
   \]

2. Once \(\gamma\) is known, \(\beta\) can be calculated using the fact that the angles of a triangle sum up to \(180^\circ\):
   \[
   \beta = 180^\circ - \alpha - \gamma
   \]

3. Finally, use the Law of Sines again to find \(b\):
   \[
   b = \frac{a \cdot \sin(\beta)}{\sin(\alpha)}
   \]

---

#### b) Given: \(a = 8.2 \, \text{cm}, b = 4.5 \, \text{cm}, \beta = 18^\circ\)

1. First, find angle \(\alpha\) using the Law of Sines:
   \[
   \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)}
   \]
   Solve for \(\sin(\alpha)\):
   \[
   \sin(\alpha) = \frac{a \cdot \sin(\beta)}{b}
   \]

2. Once \(\alpha\) is known, calculate \(\gamma\):
   \[
   \gamma = 180^\circ - \alpha - \beta
   \]

3. Finally, use the Law of Sines to find the remaining side \(c\):
   \[
   \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)}
   \]

---

#### c) Given: \(b = 6.2 \, \text{cm}, c = 3.9 \, \text{cm}, \gamma = 30^\circ\)

1. Use the Law of Sines to find \(\alpha\):
   \[
   \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}
   \]

2. After finding \(\alpha\), calculate \(\beta\):
   \[
   \beta = 180^\circ - \alpha - \gamma
   \]

3. Then use the Law of Sines again to find \(a\).

---

#### d) Given: \(a = 4.4 \, \text{cm}, c = 8.7 \, \text{cm}, \alpha = 15^\circ\)

1. Use the Law of Sines to find \(\gamma\):
   \[
   \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)}
   \]

2. Find \(\beta\) using the angle sum rule:
   \[
   \beta = 180^\circ - \alpha - \gamma
   \]

3. Finally, use the Law of Sines to find \(b\):
   \[
   \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)}
   \]

---

Would you like me to calculate the missing values for any specific sub-question? Feel free to ask! Here are some related questions you could explore:

1. What are the applications of the Law of Cosines?
2. How can you check if a triangle is obtuse or acute based on its angles?
3. When should you use the Law of Sines vs. the Law of Cosines?
4. Can the Law of Sines be used for non-right triangles?
5. How would you solve a triangle if only angles were given?

### Tip:
Remember, the Law of Sines works best when you know one angle and the opposite side. For other cases, the Law of Cosines might be more helpful.

Berechne die fehlenden Seitenlängen und Winkelgrößen im Dreieck ABC. (4 subparts with various side lengths and angles provided)

This problem focuses on calculating the missing sides and angles of triangles using the Law of Sines and Law of Cosines. It includes four different triangle setups with provided angles and sides. Learn the detailed step-by-step solution using key trigonometric formulas.

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