Answer :
To find the value of [tex]x[/tex] in the given triangle, we can use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.
In a triangle with sides [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex], and an angle [tex]\theta[/tex] between sides [tex]a[/tex] and [tex]b[/tex], the Law of Cosines is given by:
[tex]c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)[/tex]
In this case, we have:
- [tex]a = 10[/tex]
- [tex]b = 14[/tex]
- [tex]\theta = 66^\circ[/tex]
- [tex]c = x[/tex]
Plug these values into the formula:
[tex]x^2 = 10^2 + 14^2 - 2 \times 10 \times 14 \cdot \cos(66^\circ)[/tex]
Calculate each part:
- [tex]10^2 = 100[/tex]
- [tex]14^2 = 196[/tex]
- [tex]2 \times 10 \times 14 = 280[/tex]
- [tex]\cos(66^\circ) \approx 0.4067[/tex]
Substitute these into the equation:
[tex]x^2 = 100 + 196 - 280 \times 0.4067[/tex]
[tex]x^2 = 296 - 113.876[/tex]
[tex]x^2 \approx 182.124[/tex]
Now, take the square root of both sides to solve for [tex]x[/tex]:
[tex]x \approx \sqrt{182.124} \approx 13.5[/tex]
Therefore, the length of side [tex]x[/tex] is approximately 13.5 units.
