# Thread: Use the law of sines to deduce the law of tangent

1. ## Use the law of sines to deduce the law of tangent

Use the law of sines to deduce the law of tangents for triangle ABC: (a-b)/(a-b) = (tan(1/2)(A-B))/(tan(1/2)(A-B))

I've put a lot of thought into this homework problem and I hardly even know where to begin. I've attended every class and done all of the homework and none of this seems familiar to me. Any help would be much appreciated. Thanks.

a. Suppose that a, b, x, and y are real numbers such that (a/b)=(x/y), verify that (a-b)/(a+b) = (x-y)/(x+y)
b. Use the law of sines and the result in part a to show that (a-b)/(a+b)=(sinA-SinB)/((sinA+sinB)
c. Use the result in part a and the sum to product formulas to complete the derivation.

2. ## Re: Use the law of sines to deduce the law of tangent

Hi,
we use de notations ABC with alpha, betha, gamma, a,b,c
From Sinus Formula we have:
sin(alpha)/a=sin(betha)/b => b=a*sin(betha)/sin(alpha)
=>
(a+b)/(a-b)=[(a+a*sin(betha)/sin(alpha)] / )=[(a-a*sin(betha)/sin(alpha)] =
(sin(alpha)+sin(betha))/(sin(alpha)-sin(betha))
for begin.

4. ## Re: Use the law of sines to deduce the law of tangent

Yes,
it's correct.
k =2R, where R is the radius of the triangle the content the points A,B und C of the triangle(ABC is in the circle inscreibed triangle),
see sinus formulas.