Respuesta :
Answer:
AC = 6.13000744...
Step-by-step explanation:
The law of cosines is:
[tex]c^{2}[/tex] = [tex]a^{2}[/tex] + [tex]b^{2}[/tex] - 2ab(cosC)
The triangle can be drawn as attached and the numbers can be substituted into the law of cosines, switching the letters:
[tex]b^{2}[/tex] = [tex]5^{2}[/tex] + [tex]8^{2}[/tex] - 2(5)(8)(cos50º)
Then, some numbers can be simplified:
[tex]b^{2}[/tex] = 89 - 80cos50º
Finally, the square root can be taken to solve for b (side AC):
b = 6.13000744...
The law of cosine helps us to know the third side of a triangle when two sides of the triangle and an angle between the two sides are known. The length of the side AC using the law of cosine is 6.13 cm.
What is the Law of Cosine?
The law of cosine helps us to know the third side of a triangle when two sides of the triangle are already known the angle opposite to the third side is given. It is given by the formula,
[tex]c =\sqrt{a^2 + b^2 -2ab\cdot Cos\theta}[/tex]
where
c is the third side of the triangle
a and b are the other two sides of the triangle,
and θ is the angle opposite to the third side, therefore, opposite to side c.
The law of cosine can be applied in the given triangle to find the length of the side AC as shown below,
[tex]c =\sqrt{a^2 + b^2 -2ab\cdot Cos\theta}[/tex]
[tex]AC =\sqrt{(AB)^2 + (BC)^2 -2(AB)(BC)\cdot \cos(\angle B)}[/tex]
[tex]AC =\sqrt{(8)^2 + (5)^2 -2(8)(5)\cdot \cos(50^o)}[/tex]
[tex]AC =\sqrt{89 -80\cdot \cos(50^o)}[/tex]
AC = √(89 - 51.423)
AC = √(37.5769)
AC = 6.13 cm
Hence, the length of the side AC using the law of cosine is 6.13 cm.
Learn more about the Law of Cosine:
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