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in $\triangle abc$, $a = 2.6$ cm, $b = 4.3$ cm and $c = 4.7$ cm. find the measure of $\angle a$ to the nearest degree.
answer attempt 1 out of 3
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Explanation:

Step1: Recall the Law of Cosines

To find an angle in a triangle when we know all three sides, we use the Law of Cosines. The formula for angle \( A \) (opposite side \( a \)) is:
\[
\cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}
\]

Step2: Substitute the given values

We are given \( a = 2.6\space\text{cm} \), \( b = 4.3\space\text{cm} \), and \( c = 4.7\space\text{cm} \). Substitute these values into the formula:
First, calculate \( b^{2} \), \( c^{2} \), and \( a^{2} \):
\( b^{2}=4.3^{2}=18.49 \)
\( c^{2}=4.7^{2}=22.09 \)
\( a^{2}=2.6^{2}=6.76 \)
Then, substitute into the numerator: \( b^{2}+c^{2}-a^{2}=18.49 + 22.09-6.76=33.82 \)
And the denominator: \( 2bc = 2\times4.3\times4.7 = 40.42 \)
So, \( \cos A=\frac{33.82}{40.42}\approx0.8367 \)

Step3: Find the angle \( A \)

To find \( A \), take the inverse cosine (arccos) of \( 0.8367 \):
\( A=\arccos(0.8367)\approx33^{\circ} \) (rounded to the nearest degree)

Answer:

\( 33^{\circ} \)