Question

Cho tam giác ABC, góc A=90 độ, góc B=alpha. Chứng minh rằng.a, tan alpha =sin alpha/cos alpha

b, tan alpha.cot alpha=1

Answer 1

a,Có sinα=\(\dfrac{AC}{BC}\)

cosα=\(\dfrac{AB}{BC}\)

sinα/cosα=AC/BC : AB/BC=AC/AB=tanα

b,Có tanα=AC/BC

cotα=BC/AC

tanα x cotα=AC/BC x BC/AC=1

Answer 2

a)Xét \(\Delta ABC\) vuông tại A

Ta có :\(\sin\alpha=\dfrac{AC}{BC}\)

\(\cos\alpha=\dfrac{AB}{BC}\)

\(\Rightarrow\dfrac{\sin\alpha}{\cos\alpha}=\dfrac{\dfrac{AC}{BC}}{\dfrac{AB}{BC}}\)

\(\Rightarrow\dfrac{\sin\alpha}{\cos\alpha}=\dfrac{AC}{BC}.\dfrac{BC}{AB}\)

\(\Rightarrow\dfrac{\sin\alpha}{\cos\alpha}=\dfrac{AC}{AB}=\tan\alpha\)

b)Ta có: \(\tan\alpha=\dfrac{AC}{AB}\)

\(\cot\alpha=\dfrac{AB}{AC}\)

\(\Rightarrow\tan\alpha.\cot\alpha=\dfrac{AC}{AB}.\dfrac{AB}{AC}=1\)