ta có (sinA + cosA )*2 = sinA*2+cosA *2 + 2sinAcosA = 1+ 2sinAcosA > 1 .

Vì A là gọc nhọn nên sinA hay CosA > 0 ,

\(\dfrac{\left(cosa-sina\right)^2-\left(cosa+sina\right)^2}{cosa\cdot sina}\)

\(=\dfrac{\left(cosa-sina-cosa-sina\right)\left(cosa-sina+cosa+sina\right)}{cosa\cdot sina}\)

\(=\dfrac{-2\cdot sina\cdot2\cdot cosa}{cosa\cdot sina}=-4\)

Gọi tam giác vuông đề bài cho là ΔABC vuông tại A, có \(\hat{B}=\alpha\)

Xét ΔABC vuông tại A có

\(\sin\alpha=\sin B=\frac{AC}{BC};cos\alpha=cosB=\frac{AB}{BC}\)

\(\tan\alpha=\tan B=\frac{AC}{AB};\cot\alpha=\cot B=\frac{AB}{AC}\)

\(\frac{\sin\alpha}{cos\alpha}=\frac{AC}{BC}:\frac{AB}{BC}=\frac{AC}{AB}=tan\alpha\)

\(\frac{cosa}{\sin a}=\frac{cosB}{\sin B}=\frac{AB}{BC}:\frac{AC}{BC}=\frac{AB}{AC}=\cot\alpha\)

\(\tan a\cdot\cot\alpha=\frac{\sin\alpha}{cos\alpha}\cdot\frac{cos\alpha}{\sin\alpha}=1\)

\(\sin^2\alpha+cos^2\alpha=\left(\frac{AB}{BC}\right)^2+\left(\frac{AC}{BC}\right)^2=\frac{AB^2+AC^2}{BC^2}=\frac{BC^2}{BC^2}=1\)

bạn tự vẽ hình nha thông cảm cho mình

a) vẽ đường cao BH (BH⊥AC,H∈AC)

Ta có : \(\sin A+\cos A=\frac{BH}{AB}+\frac{AH}{AB}\)\(\left(\sin A=\frac{BH}{AB},\cos A=\frac{AH}{AB}\right)\)

\(\Leftrightarrow\sin A+\cos A=\frac{BH+AH}{AB}\)

Xét tam giác AHB ta có : \(BH+AH>AB\) (BĐT tam giác)

\(\Leftrightarrow\)\(\frac{BH+AH}{AB}>1\)

\(\Leftrightarrow\sin A+cosA>1\)(đpcm)

b)Ta có :\(\cot B=\frac{BH}{AH},\cot C=\frac{HC}{AH},BH+HC=BC\)

VP:\(AH\cdot\left(\cot B+\cot C\right)\)

\(=AH\cdot\left(\frac{BH}{AH}+\frac{HC}{AH}\right)\)

\(=BH+HC\)

\(=BC\) (đpcm)

c) Ta có:\(\tan B=\frac{AH}{BH}\)

Hay \(\tan\left(60\right)=\frac{6}{BH}\)

\(\Leftrightarrow BH=\frac{6}{\tan\left(60\right)}\)

\(\Leftrightarrow BH=2\sqrt{3}\)

Ta có :\(\tan\left(45\right)=\frac{AH}{HC}\)

Hay \(\tan\left(45\right)=\frac{6}{HC}\)

\(\Leftrightarrow HC=\frac{6}{\tan\left(45\right)}\)

\(\Leftrightarrow HC=6\)

Ta có :BH+HC=BC

Hay \(2\sqrt{3}+6=BC\)

\(\Leftrightarrow2\sqrt{3}+6\approx9.5\)

Ta có: SABC \(=\frac{1}{2}\cdot BC\cdot AH\)

Hay SABC\(=\frac{1}{2}6\cdot9.5\)

\(\Leftrightarrow SABC=28.5\)

Vậy SABC=28.5cm

mình nhầm \(28.5cm^2\)

Câu hỏi của Ngô Hà Minh - Toán lớp 9 - Học toán với OnlineMath

d: \(\sin B\cdot cosC+\sin C\cdot cosB\)

\(=\sin\left(B+C\right)\)

\(=\sin\left(180^0-A\right)=\sin A\)

e: tan A+tan B+tan C

\(=\tan\left(A+B\right)\left(1-\tan A\cdot\tan B\right)+\tan C\)

\(=\tan\left(180^0-C\right)\left(1-\tan A\cdot\tan B\right)+\tan C\)

\(=-\tan C\left(1-\tan A\cdot\tan B\right)+\tan C\)

\(=-\tan C+\tan A\cdot\tan B\cdot\tan C+\tan C=\tan A\cdot\tan B\cdot\tan C\)

<=> sin²A +cos²A +2sinA.cosA>1 ( do sinA + cosA >0)

<=> 1+ 2sinA.cosA >1

Dieu nay luon dung do 2sinA.cosA >0 voi moi gia tri cua A

Study well

\(\cos a-\sin a=\dfrac{1}{5}\\ \Leftrightarrow\left(\cos a-\sin a\right)^2=\dfrac{1}{25}\\ \Leftrightarrow1-2\sin a\cos a=\dfrac{1}{25}\\ \Leftrightarrow2\sin a\cos a=\dfrac{24}{25}\)

Mà \(\cos a=\dfrac{1}{5}+\sin a\)

\(\Leftrightarrow2\sin a\left(\dfrac{1}{5}+\sin a\right)=\dfrac{24}{25}\\ \Leftrightarrow\dfrac{2}{5}\sin a+2\sin^2a-\dfrac{24}{25}=0\\ \Leftrightarrow\left[{}\begin{matrix}\sin a=\dfrac{3}{5}\\\sin a=-\dfrac{4}{5}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\cos a=\dfrac{4}{5}\\\cos a=-\dfrac{3}{5}\end{matrix}\right.\\ \Leftrightarrow\cot a=\dfrac{4}{5}\cdot\dfrac{5}{3}=\dfrac{4}{3}\)