To find the angle $A$ in a triangle given all three sides $a$, $b$, and $c$, we can use the Law of Cosines. The formula for angle $A$ is:

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Where:

• $a = 100 \, \text{cm}$
• $b = 82.46 \, \text{cm}$
• $c = 63.25 \, \text{cm}$

Let's plug in the values:

$\cos A = \frac{(82.46)^2 + (63.25)^2 - (100)^2}{2 \times 82.46 \times 63.25}$

First, calculate the squared values and the numerator:

$82.46^2 = 6800.24$ $63.25^2 = 3999.56$ $100^2 = 10000$ $b^2 + c^2 - a^2 = 6800.24 + 3999.56 - 10000 = 799.8$

Now calculate the denominator:

$2 \times 82.46 \times 63.25 = 10437.45$

So,

$\cos A = \frac{799.8}{10437.45} \approx 0.0766$

Now, find the angle $A$ by taking the inverse cosine (arccos) of 0.0766:

$A = \cos^{-1}(0.0766) \approx 85.60^\circ$

Thus, angle $A$ is approximately 85.60 degrees.

Would you like further details or have any questions about the process?

Related Questions:

1. How can you calculate the remaining angles $B$ and $C$ in triangle ABC?
2. What is the Law of Cosines, and how can it be applied to other scenarios?
3. What would happen to the angle $A$ if side $a$ were shorter?
4. How does the Law of Cosines compare to the Law of Sines?
5. Can the Law of Cosines be used for non-triangle shapes like quadrilaterals?

Tip: The Law of Cosines is particularly useful for non-right triangles where none of the angles are known.