. In triangle ABC, b = 64.5, c = 38.1 and A = 58°34′
. Calculate the value of a.

To calculate the side \( a \) in triangle \( ABC \) given that \( b = 64.5 \), \( c = 38.1 \), and \( A = 58^\circ34' \), we can use the **Law of Cosines** again. The Law of Cosines states:

\[
a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
\]

**Step 1: Convert the angle to decimal degrees**

First, we convert \( A = 58^\circ34' \) to decimal degrees:

\[
58^\circ + \frac{34'}{60} = 58^\circ + 0.5667^\circ = 58.5667^\circ
\]

**Step 2: Apply the Law of Cosines**

Now, we substitute the known values into the Law of Cosines:

\[
a^2 = 64.5^2 + 38.1^2 - 2 \times 64.5 \times 38.1 \times \cos(58.5667^\circ)
\]

**Step 3: Calculate the squares of \( b \) and \( c \)**

\[
64.5^2 = 4160.25
\]
\[
38.1^2 = 1451.61
\]

**Step 4: Calculate the cosine term**

\[
\cos(58.5667^\circ) \approx 0.5250
\]
\[
2 \times 64.5 \times 38.1 \times 0.5250 \approx 2569.99
\]

**Step 5: Substitute the values into the equation**

\[
a^2 = 4160.25 + 1451.61 - 2569.99
\]
\[
a^2 = 3041.87
\]
\[
a = \sqrt{3041.87} \approx 55.18
\]

So, the side \( a \) is approximately \( 55.18 \) units.

Would you like more details or have any questions?

Here are some related questions:

1. How would you calculate the other angles of triangle ABC?
2. What is the importance of converting angles from degrees and minutes to decimal form in trigonometry?
3. Can you find the area of triangle ABC with the given information?
4. How would you verify the result using the sine rule?
5. What are some practical applications of the Law of Cosines in real life?

**Tip:** Always remember to convert angles given in degrees and minutes to decimal form before using trigonometric functions. This helps in avoiding calculation errors.

Learn how to calculate side 'a' in triangle ABC using the Law of Cosines with provided angles and side lengths. This problem involves trigonometric concepts, including angle conversion to decimal degrees, and practical applications of the Law of Cosines. Suitable for high school students.

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