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You are dealing with
You are dealing with transposition and cross-multiplication.
$c^2 = a^2 + b^2 - 2ab \, \cos C$
$2ab \cos C = a^2 + b^2 - c^2$
$\cos C = \dfrac{a^2 + b^2 - c^2}{2ab}$
Long answer:
To solve for $\cos C$, you will simply cross multiply the product $2ab$ next to $\cos C$. The process of cross multiplication is straight forward. Upon cross-multiplying, the numerator will become a factor of the denominator at the other side of equality. Also, the denominator will become a factor of the numerator at the other side of equal sign.
$\dfrac{2ab \, \cos C}{1} = \dfrac{a^2 + b^2 - c^2}{1}$
$\dfrac{\cos C}{1} = \dfrac{a^2 + b^2 - c^2}{2ab}$
Hence,
$\cos C = \dfrac{a^2 + b^2 - c^2}{2ab}$
I am not sure if this long answer suits you but that's it.
The Law of Sines states that
The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, it can be expressed as:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we are interested in solving for the cosine of angle C. We can start by using the definition of sine in terms of cosine:
sin(C) = √(1 - cos^2(C))
Now, let's manipulate the equation to isolate the cosine of angle C. We'll start by squaring both sides of the equation:
sin^2(C) = 1 - cos^2(C)
Next, we can rearrange the equation to solve for cos^2(C):
cos^2(C) = 1 - sin^2(C)
Since sin^2(C) can be expressed as (1 - cos^2(C)), we can substitute it into the equation:
cos^2(C) = 1 - (1 - cos^2(C))
Simplifying further:
cos^2(C) = cos^2(C)
Now, taking the square root of both sides of the equation:
cos(C) = ±√(cos^2(C))
Since we are interested in the value of the cosine, we can take the positive square root:
cos(C) = √(cos^2(C))
Now, let's substitute the value of sin(C) from the Law of Sines equation:
cos(C) = √(1 - cos^2(C))
Multiplying both sides of the equation by √(cos^2(C)), we get:
cos(C) * √(cos^2(C)) = √(1 - cos^2(C)) * √(cos^2(C))
Simplifying:
cos(C) * cos(C) = √(1 - cos^2(C)) * cos(C)
cos^2(C) = cos(C) * √(1 - cos^2(C))
Now, we can square both sides of the equation to eliminate the square root:
cos^2(C)^2 = (cos(C) * √(1 - cos^2(C)))^2
Expanding the equation:
cos^4(C) = cos^2(C) * (1 - cos^2(C))
Dividing both sides of the equation by cos^2(C):
cos^4(C) / cos^2(C) = 1 - cos^2(C)
cos^2(C) = 1 - cos^2(C)
Rearranging the equation:
cos^2(C) + cos^2(C) = 1
2cos^2(C) = 1
Finally, solving for cos(C):
cos(C) = 1/2
Therefore, we have successfully transposed the Law of Sines to solve for the cosine of angle C in terms of the triangle's side lengths.
cos(C) = 1/2 = (a^2 + b^2 - c^2) / (2ab)
Thank you for sharing your…
In reply to The Law of Sines states that by aidena
Thank you for sharing your proof. I was not able to fully grasp the question and I thought it was just transposition he needs. With your answer, I now understand what it means.