Triangle Cosine Value Calculation using Law of Cosines
This solution demonstrates how to find the value of $\cos A$ in Triangle ABC given the lengths of its sides: $a = 25$, $b = 45$, and $c = 30$.
Applying the Law of Cosines
To find the cosine of angle A when all three side lengths are known, we use the Law of Cosines:
$ \cos A = \frac{b^2 + c^2 - a^2}{2bc} $
Substituting Given Values
Substitute the provided side lengths into the formula:
- $a = 25$
- $b = 45$
- $c = 30$
$ \cos A = \frac{45^2 + 30^2 - 25^2}{2 \times 45 \times 30} $
Calculation Steps
- Calculate the square of each side length:
- $45^2 = 2025$
- $30^2 = 900$
- $25^2 = 625$
- Calculate the numerator of the fraction: $2025 + 900 - 625 = 2300$.
- Calculate the denominator of the fraction: $2 \times 45 \times 30 = 2700$.
- Form the fraction for $\cos A$: $\cos A = \frac{2300}{2700}$.
- Simplify the fraction by dividing both the numerator and denominator by 100: $ \cos A = \frac{23}{27} $
Conclusion
The value of $\cos A$ for Triangle ABC is $\frac{23}{27}$.
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