I saw on a website, law of sines are transposed. Example:
c^2=a^2+b^2-2abcos(C)
Transposed: cos(C)= a^2+b^2-c^2/2ab
I need a proof how it became this way.
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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 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)
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