a) sin anpha = 2/3 => góc anpha = 42^o 

cos 42^o = 0,743

tan 42^o =  0,9

cot  42^o = 1/tan 42^o = 1/0,9 = 1,111

b) tan anpha + cot anpha = 3

<=> tan anpha + 1/tan anpha = 3

<=> tan²  anpha = 2

<=> tan anpha = \(\sqrt{2}\)

=> góc anpha =  55^o 

Ta có: a = sin 55^o . cos 55^o

<=> a = 0,469

\(\sin\alpha=\frac{2}{5}\)

\(\Rightarrow\cos\alpha=\sqrt{1-\sin^2\alpha}\)

\(=\sqrt{1-\frac{4}{25}}\)

\(=\sqrt{\frac{21}{25}}=\)\(\frac{\sqrt{21}}{5}\)

\(\Rightarrow\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=\frac{2}{5}:\frac{\sqrt{21}}{5}=\frac{2}{\sqrt{21}}\)và \(\cot\alpha=\frac{\sqrt{21}}{2}\)

2. Tương tự a)

\(\cos B=\sqrt{1-\sin^2B}\)

\(=\sqrt{1-\frac{1}{4}}\)

\(=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}\)

\(\tan B,\cot B\)bạn tự tính nốt.

\(sin\alpha=0,4\Rightarrow sin^2\alpha=0,16\Rightarrow cos^2\alpha=1-sin^2\alpha=1-0,16=0,84\Rightarrow cos\alpha=\frac{\sqrt{21}}{5}\)

\(tan\alpha=\frac{sin\alpha}{cos\alpha}=\frac{0,4}{\frac{\sqrt{21}}{5}}=\frac{2\sqrt{21}}{21}\)

\(cot\alpha=1:sin\alpha=1:\frac{2\sqrt{21}}{21}=\frac{21}{2\sqrt{21}}\)

\(0< a< \frac{\pi}{2}\Rightarrow sina;cosa;tana>0\)

\(tana+\frac{1}{tana}=3\Leftrightarrow tan^2a-3tana+1=0\) \(\Rightarrow\left[{}\begin{matrix}tana=\frac{3-\sqrt{5}}{2}\\tana=\frac{3+\sqrt{5}}{2}\end{matrix}\right.\)

- Với \(tana=\frac{3-\sqrt{5}}{2}\)

\(\Rightarrow cota=\frac{1}{tana}=\frac{3+\sqrt{5}}{2}\)

\(1+tan^2a=\frac{1}{cos^2a}\Rightarrow cosa=\frac{1}{\sqrt{1+tan^2a}}=\frac{2}{\sqrt{18-6\sqrt{5}}}\)

\(sina=\sqrt{1-cos^2a}=\frac{2}{\sqrt{18+6\sqrt{5}}}\)

\(cos\left(\frac{3\pi}{2}-a\right)=cos\left(2\pi-\frac{\pi}{2}-a\right)=-sina=...\)

\(sin\left(2\pi+a\right)=sina=...\)

\(tan\left(\pi-a\right)=-tana=...\)

\(cot\left(\pi+a\right)=cota=...\)

TH2: \(tana=\frac{3+\sqrt{5}}{2}\)

Tương tự như trên

\(cot^2a=\left(\dfrac{a^2-b^2}{2ab}\right)^2\Leftrightarrow\dfrac{cos^2a}{sin^2a}=\dfrac{a^4+b^4-2a^2b^2}{4a^2b^2}\)

\(\Leftrightarrow\dfrac{cos^2a}{sin^2a}+1=\dfrac{a^4+b^4-2a^2b^2}{4a^2b^2}+1\)

\(\Leftrightarrow\dfrac{1}{sin^2a}=\dfrac{a^4+b^4+2a^2b^2}{4a^2b^2}\)

\(\Leftrightarrow sin^2a=\dfrac{4a^2b^2}{a^4+b^4+2a^2b^2}\)

\(\Leftrightarrow cos^2a=1-sin^2a=1-\dfrac{4a^2b^2}{a^4+b^4+2a^2b^2}=\dfrac{a^4+b^4-2a^2b^2}{a^4+b^4+2a^2b^2}\)

\(\Leftrightarrow cos^2a=\left(\dfrac{a^2-b^2}{a^2+b^2}\right)^2\)

\(\Leftrightarrow cosa=\dfrac{a^2-b^2}{a^2+b^2}\)

\(1+tan^2a=\dfrac{1}{cos^2a}=\dfrac{1}{\dfrac{1}{25}}=25\)

=>tan²a=24

hay \(tana=2\sqrt{6}\)

=>cot a=căn 6/12