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We stat with the law of cosines, which we can assume to be true:
* c2 = a2 + b2 - 2ab*cos(C) Then rearrange it:
* cos(C) = a2 + b2 - c2/2ab Use the identity sin(x)=SQRT(1-cos(x))
* sin(C) = SQRT( 1 - (a2 + b2 - c2/2ab)2) Use the operator A = 1/2ab*sin(C) where A is area. Also, set one equal to 4a2b2 and factor it out.
* 2A/ab = SQRT(4a2b2 - (a2 + b2 - c2)2)/2ab ab's cancel, and the term inside the square root is the difference of two squares.
* A = 1/4*SQRT((2ab - (a2 + b2 - c2))(2ab + (a2 + b2 - c2))) when the two groups are simplified, the can be factored in binomial squares.
* A = 1/4*SQRT((c2 - (a - b)2)((a + b)2 - c2) Once again, we have differences of squares.
* A = 1/4*SQRT((c - (a - b))(c + (a - b))((a + b) - c)((a + b + c)) Simplify.
* A = 1/4*SQRT((c + b - a)(c + a - b)(a + b - c)(a + b +c)) Here comes the tricky part. We have four parts here. Three have two terms positive and one negative. Having a + b + c is like having the P. If we have a + b - c, that is like saying P - 2c, right? So to make it even easier, we can call s, the semi-perimeter, P/2. Then we can say a + b - c is 2s - 2c, or 2(s - c). We can apply that to all parts except the last one, which is just 2s.
* A = 1/4*SQRT(2(s - a)*2(s - b)*2(s - c)*2s) The two's can multiply together to 16 and come out of the root, canceling with the 1/4. we are left with good old Heron's formula.
* A = SQRT(s(s - a)(s - b)(s - c))
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Who invented the cosine rule?
It was invented sometim but if you find out put it on here! Love Megan
What formula do you use to find the perimerter of a SSA triangle?
An SSA triangle is ambiguous.Suppose the triangle is ABC and, with conventional labelling, you know a, b and angle A.Then by the cosine rule, a2 = b2 + c2 - 2bc*Cos(A)This equation will give rise to a quadratic equation in cwhich has 2 solutions. The perimeter is then a + b + c1 or a + b + c2
When doing trigonometry for example the cosine rule how do you know which side is a which side is b and which side is c if none of them are marked on and neither are the angles?
It's possible that either the angles or sides are labeled according to length or size.
In the Sine and Cosine rule does it matter where the abc ABC go on the triangle for example can it go in any order or must it go in alphabetically order?
The order of them does not matter at all, as long as the sides are consistently opposite the angles with the corresponding letter (e.g. side "A" is always opposite angle "a").
A triangular plot of land has sides that measure 5 meters 7 meters and 10 meters What is the area of this plot of land to the nearest tenth of a square meter?
This is NOT a Pythagorean triangle. So we must fall back on the Cosine Rule for an agular value. Cosine Rule is a^(2) = b^(2) + c^(2) - 2bcCosA Algebraically rearranging CosA = [a^(2) - b^(2) - c^(2)] / -2bc Substituting CosA = [10^(2) - 7^(2) - 5^)2)] / -(2(7)(5)) CosA = [100 - 49 - 25]/-70 CosA = 26/-70 CosA = -0.371428571... A = Cos^(-1) -0.371428573 A =111.8037 481 ... degrees. - Now area of triangle is 0.5 X Base X height The height is = 7Sin(111.8037481...) Hence are is 0.5(5)(7)Sin(111.8037481...) Area = 16.248... m^(2)
