Trigonometry Preparing for math contest; Please help

[MathExpert]

Prove that the triangle ABC is:

a) acute, if and only if tan(alfa)*tan(beta)>1
b) right, if and only if tan(alfa)*tan(beta)=1
c) obtuse, if and only if tan(alfa)*tan(beta)<1

 

[tongos]

start with the following...
k=alpha+beta
90<k<180 in an acute triangle
k=90 in a right triangle
k<90 in an obtuse triangle

(sinx/cosx)(sin(k-x)/cos(k-x))
=(sinxsinkcosx-cosksin^2x)/(cos^2xcosk+sinksinxcosx)
sinxsinkcosx=A and cosk=B
(A-Bsin^2x)/(A+Bcos^2x)= (A-Bsin^2x)/(A+B-Bsin^2x)
we then want B equal to zero. cosk=0 when k=90. when k is greater B becomes negative.

 

[tacman]

Before we begin, it is important to note that the triangle ABC has three angles, alfa, beta, and gamma, and three sides, a, b, and c. The angles alfa and beta are adjacent to the side c, while the angle gamma is opposite the side c. Additionally, we will be using the trigonometric ratio tangent (tan) in our proofs.

a) To prove that triangle ABC is acute if and only if tan(alfa)*tan(beta)>1, we will first assume that the triangle is acute. This means that all three angles are less than 90 degrees. Using the tangent ratio, we know that tan(alfa)=a/c and tan(beta)=b/c. Multiplying these two ratios together, we get: tan(alfa)*tan(beta)=(a/c)*(b/c)=(ab)/(c^2). Since all three angles are less than 90 degrees, the side c must be the longest side according to the triangle inequality theorem. This means that c^2>a^2 and c^2>b^2. Substituting this into our equation, we get: (ab)/(c^2)<1. Therefore, tan(alfa)*tan(beta)<1.

Now, assuming that tan(alfa)*tan(beta)>1, we can prove that the triangle ABC is acute. If tan(alfa)*tan(beta)>1, then (ab)/(c^2)>1. This means that ab>c^2, and by the triangle inequality theorem, we know that c must be the longest side. Therefore, all three angles must be less than 90 degrees, making the triangle ABC acute.

b) Similarly, to prove that the triangle ABC is right if and only if tan(alfa)*tan(beta)=1, we will first assume that the triangle is right. This means that one angle is equal to 90 degrees. Without loss of generality, let's assume that angle alfa is the right angle. Using the tangent ratio, we know that tan(alfa)=a/c and tan(beta)=b/c. Substituting these into our equation, we get: tan(alfa)*tan(beta)=(a/c)*(b/c)=(ab)/(c^2)=1. Therefore, tan(alfa)*tan(beta)=1.

Conversely, assuming that tan(alfa)*tan(beta)=1, we can prove that the triangle ABC is right. If tan(alfa)*tan(beta)=1, then