two sides and an angle are given determine whether the given measurements produce one triangle two triangles or no triangle at all solve each triangle that results side a =13 side b= 14 angle a = 62

To determine the number of triangles formed, we need to apply the **Law of Sines** and analyze the given measurements.

### Step 1: Determine if we have one, two, or no triangle
1. **Given Information:**
   - Side \( a = 13 \)
   - Side \( b = 14 \)
   - Angle \( A = 62^\circ \)

2. **Using Law of Sines:**
   \[
   \frac{\sin B}{b} = \frac{\sin A}{a}
   \]

   Substitute the values:
   \[
   \frac{\sin B}{14} = \frac{\sin 62^\circ}{13}
   \]

3. Solve for \( \sin B \):
   \[
   \sin B = \frac{14 \cdot \sin 62^\circ}{13}
   \]
   Calculating this value:
   \[
   \sin B \approx \frac{14 \cdot 0.8829}{13} \approx 0.951
   \]

   Since \( \sin B \approx 0.951 \leq 1 \), there is at least one solution for \( B \), so we proceed to find the angle.

4. **Finding Angle \( B \):**
   \[
   B \approx \sin^{-1}(0.951) \approx 72.6^\circ
   \]

### Step 2: Checking for Two Possible Triangles
Since \( \sin B = 0.951 \), there is also a possible second angle \( B' \) where:
\[
B' = 180^\circ - 72.6^\circ \approx 107.4^\circ
\]

Now, check if this configuration is possible with the given angle \( A \):

- If we use \( B = 107.4^\circ \):
  \[
  A + B' = 62^\circ + 107.4^\circ = 169.4^\circ
  \]
  Since the sum is less than \( 180^\circ \), a second triangle is possible.

### Step 3: Solve Each Triangle

#### Triangle 1 (with \( B \approx 72.6^\circ \)):
1. **Calculate \( C \):**
   \[
   C = 180^\circ - (A + B) = 180^\circ - (62^\circ + 72.6^\circ) \approx 45.4^\circ
   \]

2. **Calculate Side \( c \) using Law of Sines:**
   \[
   \frac{c}{\sin C} = \frac{a}{\sin A}
   \]
   \[
   c = \frac{13 \cdot \sin 45.4^\circ}{\sin 62^\circ}
   \]
   \[
   c \approx \frac{13 \cdot 0.7114}{0.8829} \approx 10.48
   \]

   **Solution for Triangle 1:**
   - \( A = 62^\circ \), \( B \approx 72.6^\circ \), \( C \approx 45.4^\circ \)
   - \( a = 13 \), \( b = 14 \), \( c \approx 10.48 \)

#### Triangle 2 (with \( B' \approx 107.4^\circ \)):
1. **Calculate \( C' \):**
   \[
   C' = 180^\circ - (A + B') = 180^\circ - (62^\circ + 107.4^\circ) \approx 10.6^\circ
   \]

2. **Calculate Side \( c' \) using Law of Sines:**
   \[
   c' = \frac{13 \cdot \sin 10.6^\circ}{\sin 62^\circ}
   \]
   \[
   c' \approx \frac{13 \cdot 0.184} {0.8829} \approx 2.71
   \]

   **Solution for Triangle 2:**
   - \( A = 62^\circ \), \( B' \approx 107.4^\circ \), \( C' \approx 10.6^\circ \)
   - \( a = 13 \), \( b = 14 \), \( c' \approx 2.71 \)

### Conclusion
Two triangles are possible with the given measurements:

1. **Triangle 1:**
   - \( A = 62^\circ \), \( B \approx 72.6^\circ \), \( C \approx 45.4^\circ \)
   - \( a = 13 \), \( b = 14 \), \( c \approx 10.48 \)

2. **Triangle 2:**
   - \( A = 62^\circ \), \( B' \approx 107.4^\circ \), \( C' \approx 10.6^\circ \)
   - \( a = 13 \), \( b = 14 \), \( c' \approx 2.71 \)

Would you like more details on any steps? Here are some related questions to explore:

1. How can the Law of Sines help determine ambiguous cases?
2. Why does the sine function allow for two possible angles?
3. What conditions guarantee only one triangle exists?
4. How can the Law of Cosines verify triangle solutions?
5. What real-world scenarios involve ambiguous triangle cases?

**Tip:** When two triangles are possible, always check the sum of angles to ensure a valid triangle.

Two sides and an angle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Side a = 13, side b = 14, angle A = 62°.

Explore how to determine if one, two, or no triangles can be formed when two sides and an angle are given, with side a = 13, side b = 14, and angle A = 62°. This trigonometry problem illustrates the ambiguous case in the Law of Sines, showing how to calculate possible angles and sides for each triangle.

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