To find the cosine of an angle in a triangle using the law of cosines, we can use the following formula:
\[ \cos(\theta) = \frac{b^2 + c^2 - a^2}{2bc} \]
Where:
- \(a\), \(b\), and \(c\) are the lengths of the sides opposite to angles \(A\), \(B\), and \(C\) respectively.
- \(\theta\) is the angle opposite to side \(a\).
Given:
- Side \(a = 10.2\)
- Side \(b = 5.7\)
- Side \(c = 9.8\)
We are interested in finding \(\cos(A)\), where \(A\) is opposite side \(a\).
Using the formula:
\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \]
Now, let's calculate each part:
-
Calculate \(b^2\): \[ b^2 = (5.7)^2 = 32.49 \]
-
Calculate \(c^2\): \[ c^2 = (9.8)^2 = 96.04 \]
-
Calculate \(a^2\): \[ a^2 = (10.2)^2 = 104.04 \]
Now plug these values into the cosine formula:
\[ \cos(A) = \frac{32.49 + 96.04 - 104.04}{2 \cdot 5.7 \cdot 9.8} \]
-
Calculate the numerator: \[ 32.49 + 96.04 - 104.04 = 24.49 \]
-
Calculate the denominator: \[ 2 \cdot 5.7 \cdot 9.8 = 112.08 \]
Now, perform the division:
\[ \cos(A) = \frac{24.49}{112.08} \approx 0.218 \]
Finally, rounding to two decimal places:
\[ \cos(A) \approx 0.22 \]
Thus, the value of \(\cos(A)\) to two decimal places is 0.22.