Question

how can we find trignometric ratios of 45°,60°and 30° geometrically

Answer 1

In right angle triangle Δ ABC ∠B = 90°, AB = BC = a , ∠C = 45°
AC ² = a² + a² 
=2a² 
AC = √2a²
sin 45° = opp/ hyp = a / √2a  ( a will be cancelled ) 
sin 45°= 1/√2
cos 45° = adj/ hyp = 1/ √2
tan 45° = 1 
cosec  45 ° = √2
sec 45° = √2
cot 45° = 1

RATIOS OF 60° AND 30°
consider an equilateral Δ ABC with side a .
ΔABC AB = AC = a, AD ⊥ BC , BD = DC = a / 2 , AD = √3/ 2, ∠C = 60°
sin 60 ° = opp / hyp = √3a/ 2 / a ( a will be cancelled )
sin 60° = √3/2
cos 60° = 1/2
tan 60° = √3
cosec 60° = 2/√3
sec 60° = 2
cot 60° = 1/√3

ratios 30° 
right angle Δ ABC ∠ A = 30° , AB = √3/2 , BC = a / 2 , AC = a
sin 30°= 1/2
cos 30°= √3/2
tan 30° = 1/ √3 
cosec 30°= 2
sec 30°= 2/√3
cot 30° = √3