Solve the triangle.
14
a
23
Upper B
Upper C
25 degrees
Upper A
Question content area bottom
Part 1
What is the length of side​ a?
aequals
  
enter your response here
​(Round to the nearest whole number as​ needed.)
What is the measure of angle​ B?
Bequals
  
enter your response heredegrees
​(Round to the nearest degree as​ needed.)
What is the measure of angle​ C?
Cequals
  
enter your response heredegrees
​(Round to the nearest degree as​ needed.)

The given information for the triangle is as follows:
- Side \( b = 14 \)
- Side \( c = 23 \)
- Angle \( A = 25^\circ \)
- Side \( a \) is unknown.
- Angle \( B \) is unknown.
- Angle \( C \) is unknown.

### Step 1: Find Side \( a \) Using the Law of Cosines

We can use the **Law of Cosines** to find the length of side \( a \):
\[
a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
\]
Substitute the given values:
\[
a^2 = 14^2 + 23^2 - 2(14)(23) \cdot \cos(25^\circ)
\]
Now calculate the right side:
\[
a^2 = 196 + 529 - 2(14)(23) \cdot \cos(25^\circ)
\]
\[
a^2 = 725 - 644 \cdot \cos(25^\circ)
\]
Using \( \cos(25^\circ) \approx 0.9063 \):
\[
a^2 = 725 - 644 \cdot 0.9063
\]
\[
a^2 = 725 - 583.26
\]
\[
a^2 = 141.74
\]
\[
a \approx \sqrt{141.74} \approx 11.9
\]
Rounding to the nearest whole number:
\[
a \approx 12
\]

### Step 2: Find Angle \( B \) Using the Law of Sines

We can now use the **Law of Sines** to find angle \( B \):
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)}
\]
Rearranging to solve for \( \sin(B) \):
\[
\sin(B) = \frac{b \cdot \sin(A)}{a}
\]
Substitute the known values:
\[
\sin(B) = \frac{14 \cdot \sin(25^\circ)}{12}
\]
Using \( \sin(25^\circ) \approx 0.4226 \):
\[
\sin(B) = \frac{14 \cdot 0.4226}{12}
\]
\[
\sin(B) = \frac{5.9164}{12} \approx 0.493
\]
Now find \( B \) by taking the inverse sine:
\[
B \approx \sin^{-1}(0.493) \approx 29.5^\circ
\]
Rounding to the nearest degree:
\[
B \approx 30^\circ
\]

### Step 3: Find Angle \( C \)

Since the sum of the angles in a triangle is \( 180^\circ \), we can find angle \( C \) as follows:
\[
C = 180^\circ - A - B
\]
Substitute the values of \( A \) and \( B \):
\[
C = 180^\circ - 25^\circ - 30^\circ
\]
\[
C = 125^\circ
\]

### Final Answers:
1. The length of side \( a \approx 12 \)
2. The measure of angle \( B \approx 30^\circ \)
3. The measure of angle \( C = 125^\circ \)

Would you like more details or have any questions?

Here are 5 related questions:
1. How can the Law of Sines be used to find side lengths in other triangles?
2. What are the conditions for using the Law of Cosines in a triangle?
3. How do you calculate the third angle in a triangle given two angles?
4. What are other methods for solving oblique triangles besides Law of Cosines and Sines?
5. How does rounding affect the accuracy of trigonometric solutions?

**Tip:** Always check if the triangle is obtuse or acute when using trigonometric functions to avoid errors.

Solve the triangle. Given side b = 14, side c = 23, angle A = 25°, find the length of side a, and the measures of angles B and C.

Learn how to solve a triangle using the Law of Cosines and Law of Sines. Given sides b = 14, c = 23, and angle A = 25°, find the length of side a and the measures of angles B and C. This problem involves key trigonometric formulas and theorems, suitable for students in Grades 10-12.

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