Solve Simple Trigo Probs: Minimum Value & Proving Sin Product

[ron_jay]

Help needed to solve these:

1)what is the minimum value of the expression 9tan^2 $$\theta$$ + 4cot^2 $$\theta$$ ?

2)Prove that sin20.sin40.sin60.sin80 =3/16

 

[berkeman]

Please show us how you would approach each of these problems. We must see your work first, in order for us to provide tutorial help. We do not furnish answers to homework and coursework questions here on the PF.

 

[pardesi]

1. $$9 \tan^{2} \theta + 4 \cot^{2} \theta =(3 \tan \theta - 2 \cot \theta)^{2}+12 \geq 12$$ with equality holding when $$|\tan \theta|=\sqrt{\frac{2}{3}}$$ sorry i didn't see berkeman's post

 

[Gib Z]

To add a little understanding to pardesi's post, he wrote the expression in the form $$(3 \tan \theta - 2\cot \theta)^2 + 12$$ because even though cot and tan don't have minimum values, squares do (in the real numbers, but that's a different matter). 12 is a constant we can't change that. We know squares are more or equal to 0. So the smallest value would be if the square was 0.

So you set 3 tan x = 2 cot x. If you can find a solution, which pardesi did, then there is a value for which it is 0. Done :)