Lung tung hả

Ta có: \(sin^2\alpha+cos^2\alpha=1\Rightarrow sin^2\alpha+\left(sin\alpha+\dfrac{1}{5}\right)^2=1\)

\(\Rightarrow25sin^2\alpha+5sin\alpha-12=0\\\Rightarrow\left(5sin\alpha-3\right)\left(5sin\alpha+4\right)=0\\ \Rightarrow\left[{}\begin{matrix}sin\alpha=\dfrac{3}{5}\Rightarrow cos\alpha=\dfrac{3}{5}+\dfrac{1}{5}=\dfrac{4}{5}\Rightarrow cot\alpha=\dfrac{4}{5}:\dfrac{3}{5}=\dfrac{4}{3}\\sin\alpha=-\dfrac{4}{5}\left(loại\right)\end{matrix}\right. \)

xét cos a - sin a = 1/5 
=> (cos a - sin a)^2 = 1/25 
<=> (cos a)^2 + (sin a)^2 - 2cosasina = 1/25 
xét (cos a)^2 + (sin a)^2 =1 => -2(cos a)(sin a) = 1/25 - 1 = -24/25 
=> (cos a)(sin a) = 12/25 
=> cos a = 12/(25.sin a) 
sau đó thay vào pt ban đầu cos a - sin a = 1/5 
<=> - (sin a)^2 -1/5sina + 12/25 =0 
Giải pt bậc 2 dc 2 nghiệm 1 am 1 dương thì bạn lấy nghiệm dương do a là góc nhọn

a: Ta có: \(\sin^2a+cos^2a=1\)

=>\(cos^2a=1-0,8^2=1-0,64=0,36=0,6^2\)

=>cosa=0,6

\(\tan a=\frac{\sin a}{cosa}=\frac{0.8}{0.6}=\frac43\)

\(\cot a=\frac{1}{\tan a}=1:\frac43=\frac34\)

b: Ta có: \(\sin^2a+cos^2a=1\)

=>\(\sin^2a=1-0,6^2=1-0,36=0,64=0,8^2\)

=>sin a=0,8

\(\tan a=\frac{\sin a}{cosa}=\frac{0.8}{0.6}=\frac43\)

\(\cot a=\frac{1}{\tan a}=1:\frac43=\frac34\)

c: \(\tan a\cdot\cot a=1\)

=>\(\cot a=\frac13\)

Ta có: \(1+\tan^2a=\frac{1}{cos^2a}\)

=>\(\frac{1}{cos^2a}=1+3^2=10\)

=>\(cos^2a=\frac{1}{10}\)

=>\(cosa=\frac{1}{\sqrt{10}}\)

Ta có: \(\tan a=\frac{\sin a}{cosa}\)

=>\(\sin a=\tan a\cdot cosa\)

\(=3\cdot\frac{1}{\sqrt{10}}=\frac{3}{\sqrt{10}}\)

d: \(\tan a\cdot\cot a=1\)

=>\(\tan a\cdot2=1\)

=>\(\tan a=\frac12\)

Ta có: \(1+\tan^2a=\frac{1}{cos^2a}\)

=>\(\frac{1}{cos^2a}=1+\left(\frac12\right)^2=\frac54\)

=>\(cos^2a=\frac45\)

=>\(cosa=\frac{2}{\sqrt5}\)

Ta có: \(\tan a=\frac{\sin a}{cosa}\)

=>\(\sin a=cosa\cdot\tan a=\frac{2}{\sqrt5}\cdot\frac12=\frac{1}{\sqrt5}\)

Bài 2: 

\(\cos\widehat{A}=\dfrac{3\sqrt{39}}{20}\)

\(\tan\widehat{A}=\dfrac{7}{20}:\dfrac{3\sqrt{39}}{20}=\dfrac{7}{3\sqrt{39}}=\dfrac{7\sqrt{39}}{117}\)

\(\cot\widehat{A}=\dfrac{3\sqrt{39}}{7}\)

tana = 3/4.
=>cota=1/ tana =1:3/4=4/3
sina /cosa =tana
=> sina =tana .cosa =3/4. cosa
lại có sin^2(a)+cos^2(a)=1
<=>9/16cos^2(a)+cos^2=1
<=>25/16cos^2(a)=1
<=>cos^2(a)=16/25
=>[cosa =4/5=>sina =3/5
    [cosa =-4/5=> sina =-2/5

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

Xét ΔABC vuông tại A có

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

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

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

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

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

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

\(\sin^2\widehat{A}+\cos^2\widehat{A}=1\Leftrightarrow\cos^2\widehat{A}=1-\left(\dfrac{3}{5}\right)^2=1-\dfrac{9}{25}=\dfrac{16}{25}\\ \Leftrightarrow\cos\widehat{A}=\dfrac{4}{5}\\ \tan\widehat{A}=\dfrac{\sin\widehat{A}}{\cos\widehat{A}}=\dfrac{3}{4}\\ \Rightarrow\cot\widehat{A}=\dfrac{1}{\tan\widehat{A}}=\dfrac{4}{3}\)