Đáp án:
$1) \left[\begin{array}{l} \tan \alpha=\dfrac{3}{4}\\\tan \alpha=\dfrac{4}{3}\end{array} \right.\\ 2)\dfrac{337}{625}\\ 3)\dfrac{12}{25}\\ 4)\dfrac{91}{125}$
Giải thích các bước giải:
$3)\\ \sin \alpha +\cos \alpha=\dfrac{7}{5}(1)\\ \Leftrightarrow \left(\sin \alpha +\cos \alpha\right)^2=\left(\dfrac{7}{5}\right)^2\\ \Leftrightarrow \sin^2\alpha +2\sin \alpha \cos \alpha+\cos^2\alpha=\dfrac{49}{25}\\ \Leftrightarrow 1 +2\sin \alpha \cos \alpha=\dfrac{49}{25}\\ \Leftrightarrow 2\sin \alpha \cos \alpha=\dfrac{24}{25}\\ \Leftrightarrow \sin \alpha \cos \alpha=\dfrac{12}{25}\\ \Leftrightarrow \cos \alpha=\dfrac{12}{25\sin \alpha}\\ \text{Thế vào (1)}: \sin \alpha +\dfrac{12}{25\sin \alpha}=\dfrac{7}{5}\\ \Leftrightarrow \sin^2\alpha +\dfrac{12}{25}=\dfrac{7}{5} \sin \alpha \\ \Leftrightarrow \sin^2\alpha - \dfrac{7}{5} \sin \alpha+\dfrac{12}{25}=0\\ \Leftrightarrow \left[\begin{array}{l} \sin \alpha=\dfrac{4}{5} \Rightarrow \cos \alpha =\dfrac{3}{5}\\ \sin \alpha=\dfrac{3}{5} \Rightarrow \cos \alpha =\dfrac{4}{5}\end{array} \right.\\ 1)\tan \alpha=\dfrac{\sin \alpha}{\cos \alpha}\\ \Leftrightarrow \Leftrightarrow \left[\begin{array}{l} \tan \alpha=\dfrac{\dfrac{3}{5}}{\dfrac{4}{5}}\\\tan \alpha=\dfrac{\dfrac{4}{5}}{\dfrac{3}{5}}\end{array} \right.\\ \Leftrightarrow \left[\begin{array}{l} \tan \alpha=\dfrac{3}{4}\\\tan \alpha=\dfrac{4}{3}\end{array} \right.\\ 2)\sin^4\alpha +\cos^4\alpha\\ =\left(\dfrac{4}{5} \right)^4+\left(\dfrac{3}{5}\right)^4\\ =\dfrac{337}{625}\\ 3)\sin \alpha .\cos \alpha\\ =\dfrac{4}{5}.\dfrac{3}{5}\\ =\dfrac{12}{25}\\ 4)\sin^3\alpha +\cos^3\alpha\\ =\left(\dfrac{4}{5} \right)^3+\left(\dfrac{3}{5}\right)^3\\ =\dfrac{91}{125}$
