解：?$(1)$?在?$Rt△BMN$?中，∵?$BN=3，$??$MN=4$?
∴?$BM=\sqrt {BN^2+MN^2}=5$?
∴?$sin B=\frac {MN}{BM}=\frac 45，$??$cos B=\frac {BN}{BM}=\frac 35，$??$tan B=\frac {MN}{BN}=\frac 43$?
?$(2)$?∵?$MN⊥AB$?
∴?$∠MNB=∠C=90°$?
∴?$∠B+∠BMN=∠B+∠A=90°$?
∴?$∠BMN=∠A$?
在?$Rt△ABC$?中，∵?$AB=10，$??$BC=5$?
∴?$AC=\sqrt {AB^2-BC^2}=5\sqrt 3$?
∴?$sin ∠BMN=sin A=\frac {BC}{AB}=\frac 12，$??$cos ∠BMN=cosA=\frac {AC}{AB}=\frac {\sqrt 3}2$?
?$tan ∠BMN=tan A=\frac {BC}{AC}=\frac {\sqrt 3}3$?
?$(3)$?∵?$cos ∠BMN=cosA=\frac {AC}{AB}=\frac 34$?
不妨設(shè)?$AC=3x，$??$AB=4x$?
在?$Rt△ABC$?中，∵?$AC=3x，$??$AB=4x$?
∴?$BC=\sqrt {AB^2-AC^2}=\sqrt 7x$?
∴?$sinB=\frac {AC}{AB}=\frac {3x}{4x}=\frac 34，$??$sinA=\frac {BC}{AB}=\frac {\sqrt 7x}{4x}=\frac {\sqrt 7}4，$??$tanA=\frac {BC}{AC}=\frac {\sqrt 7x}{3x}=\frac {\sqrt 7}3$?
