Triangle ABC, a =  100cm, b = 82,46cm and c = 63,25cm. You don't know any of the angles. How many degrees is A?

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.

Triangle ABC, a = 100cm, b = 82.46cm and c = 63.25cm. You don't know any of the angles. How many degrees is A?

Learn how to calculate angle A in triangle ABC given sides a = 100 cm, b = 82.46 cm, and c = 63.25 cm using the Law of Cosines. This solution involves trigonometric concepts and the application of the cosine formula, ideal for students in Grades 9-12.

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