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Tough Trig Identity (not hw)?
Given A, B, and C are the angles of a triangle, prove:
sinC + (sinA)(sinB) ≤ (1/2)(1+sqrt(5))
I'm still trying to work it out myself so I might be late in giving an answer :P
If it helps, we know that C = pi - (A+B) and so sinC = sin(A+B) by the sum laws.
I actually figured it out, but it's still a pretty neat problem. Calculus makes it a -bit- easier, but it's very doable with only trig identities. And trig identities. You'll definately need those :P
2 Answers
- ricchettiLv 44 years ago
ok, that's the form you do it with calculus. enable's say a= first huge form b=2nd numbers a + b = 20 a^2+b^2 = some huge form now you therapy the 1st equation for a or b, that is no longer important a = 20 - b and you replace (20-b)^2+b^2= some huge form now you opperate (b^2-40*b+4 hundred)+b^2 = 2*b^2-40*b+4 hundred you could desire to now do 2 subject concerns, you could desire to graph or divide, in case you graph, then graph the function f(b)=2*b^2-40*b+4 hundred now in case you derive you get f'(b)= 4*b-40 and you're making it equatl to 0 4*b-40 = 0 and you therapy for b b = 40/4 or b = 10 and you could desire to now get a a = 20-b a = 20-10 = 10 so then you relatively particularly get that a = 10 and b = 10 that supplies you you with the minimum you like and you do the comparable with huge form 2 a + b = 30 a^2+b^2 = some huge form now you therapy the 1st equation for a or b, that is no longer important a = 30 - b and you replace (30-b)^2+b^2= some huge form now you opperate (b^2-60*b+900)+b^2 = 2*b^2-60*b+900 you could desire to now do 2 subject concerns, you could desire to graph or divide, in case you graph, then graph the function f(b)=2*b^2-60*b+900 now in case you derive you get f'(b)= 4*b-60 and you're making it such as 0 4*b-60 = 0 and you therapy for b b = 60/4 or b = 15 and you could desire to now get a a = 30-b a = 30-15 = 15 so then you relatively particularly get that a = 15 and b = 15 that supplies you you with the minimum you like desire this helps. pd. such as 0 as quickly as you derive because of the reality meaning that the slope is going to be 0 and the slope will replace into 0 together as that's optimal or minimum values.
