Certainly! Let's solve for [tex]\( c \)[/tex] step-by-step using the given values and the Law of Cosines. The Law of Cosines states:
[tex]\[
c^2 = a^2 + b^2 - 2ab \cos(\text{angle})
\][/tex]
Given values are:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 10 \)[/tex]
- [tex]\(\text{angle} = 34^\circ \)[/tex]
First, let's calculate the square of each side and the cosine term:
1. Calculate [tex]\( a^2 \)[/tex]:
[tex]\[
a^2 = 3^2 = 9
\][/tex]
2. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[
b^2 = 10^2 = 100
\][/tex]
3. Convert the angle from degrees to radians to use in the cosine function since cosine calculations typically use radians:
[tex]\[
\text{angle in radians} = \frac{34 \pi}{180} \approx 0.5934 \text{ radians}
\][/tex]
4. Calculate [tex]\(\cos(\text{angle in radians})\)[/tex]:
[tex]\[
\cos(34^\circ) \approx 0.8290
\][/tex]
5. Calculate the term involving the product of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\(\cos(\text{angle})\)[/tex]:
[tex]\[
2ab \cos(\text{angle}) = 2 \times 3 \times 10 \times 0.8290 = 49.74
\][/tex]
Now, substitute these values into the Law of Cosines formula:
[tex]\[
c^2 = 9 + 100 - 49.74
\][/tex]
Combine the terms to get:
[tex]\[
c^2 = 59.26
\][/tex]
To find [tex]\( c \)[/tex], take the square root of [tex]\( c^2 \)[/tex]:
[tex]\[
c = \sqrt{59.26} \approx 7.698
\][/tex]
Finally, round [tex]\( c \)[/tex] to the nearest tenth:
[tex]\[
c \approx 7.7
\][/tex]
So, the value of side [tex]\( c \)[/tex], rounded to the nearest tenth, is:
[tex]\[
c = 7.7
\][/tex]
