$\sin \alpha =2$?? $\sin \alpha \in [-1;1]$ với mọi $\alpha$ mà bạn. Bạn xem lại đề.

1+tan^2a=1/cos^2a

=>1/cos^2a=1+9/16=25/16

=>cos^2a=16/25

=>cosa=4/5 hoặc cosa=-4/5

a: \(\cos\alpha=\dfrac{1}{2}\)

\(\tan\alpha=\sqrt{3}\)

\(\cot\alpha=\dfrac{\sqrt{3}}{3}\)

Bài 1: 

\(\cos\alpha=\sqrt{1-\dfrac{9}{25}}=\dfrac{4}{5}\)

\(\tan\alpha=\dfrac{3}{5}:\dfrac{4}{5}=\dfrac{3}{4}\)

Bài 2: 

\(\sin\alpha=\sqrt{1-\dfrac{49}{100}}=\dfrac{\sqrt{51}}{10}\)

\(\tan\alpha=\dfrac{\sqrt{51}}{7}\)

Ta có:

\(cot\alpha\cdot tan\alpha=1\)

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

\(\Rightarrow cota=\dfrac{1}{\dfrac{3}{4}}=\dfrac{4}{3}\)

Mà:

\(cot^2\alpha+1=\dfrac{1}{sin^2\alpha}\)

\(\Rightarrow sin\alpha=\sqrt{\dfrac{1}{cot^2\alpha+1}}\)

\(\Rightarrow sin\alpha=\sqrt{\dfrac{1}{\left(\dfrac{4}{3}\right)^2+1}}=\dfrac{3}{5}\) 

Lại có:

\(cos^2\alpha+sin^2\alpha=1\)

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

\(\Rightarrow cos\alpha=\sqrt{1-\left(\dfrac{3}{5}\right)^2}=\dfrac{4}{5}\)

\(tan\alpha=\dfrac{3}{4}\\ \Rightarrow cot\alpha=1:\dfrac{3}{4}=\dfrac{4}{3}\)

Có:

\(1+cot^2\alpha=\dfrac{1}{sin^2\alpha}\\ \Rightarrow sin\alpha=\sqrt{1:\left(1+\left(\dfrac{4}{3}\right)^2\right)}=\dfrac{3}{5}\)

\(\Rightarrow cos\alpha=\sqrt{1-\left(\dfrac{3}{5}\right)^2}=\dfrac{4}{5}\)

1.a) \(4cos\dfrac{\alpha}{2}.cos\dfrac{\beta}{2}.cos\dfrac{f}{2}\)

\(=\dfrac{1}{2}.4\left[cos\left(\dfrac{\alpha-\beta}{2}\right)+cos\left(\dfrac{\alpha+\beta}{2}\right)\right].cos\dfrac{f}{2}\)

\(=2.cos\left(\dfrac{\alpha-\beta}{2}\right)cos\dfrac{f}{2}+2.cos\left(\dfrac{\alpha+\beta}{2}\right).cos\dfrac{f}{2}\)

\(=cos\left(\dfrac{\alpha-\left(\beta+f\right)}{2}\right)+cos\left(\dfrac{\alpha-\beta+f}{2}\right)+cos\left(\dfrac{\alpha+\beta-f}{2}\right)+cos\left(\dfrac{\alpha+\beta+f}{2}\right)\)

\(=cos\left(\dfrac{2\alpha-\pi}{2}\right)+cos\left(\dfrac{\pi-2\beta}{2}\right)+cos\left(\dfrac{\pi-2f}{2}\right)+cos\left(\dfrac{\pi}{2}\right)\)

\(=cos\left(-\dfrac{\pi}{2}+\alpha\right)+cos\left(\dfrac{\pi}{2}-\beta\right)+cos\left(\dfrac{\pi}{2}-f\right)\)

\(=sin\alpha+sin\beta+sinf\) (đpcm)

a2) \(1+4sin\dfrac{\alpha}{2}.sin\dfrac{\beta}{2}.sin\dfrac{f}{2}\)

\(=1+2\left[cos\left(\dfrac{\alpha-\beta}{2}\right)-cos\left(\dfrac{\alpha+\beta}{2}\right)\right].sin\dfrac{f}{2}\)

\(=1+2.cos\left(\dfrac{\alpha-\beta}{2}\right).sin\dfrac{f}{2}-2.cos\left(\dfrac{\alpha+\beta}{2}\right).sin\dfrac{f}{2}\)

\(=1+sin\left(\dfrac{f-\alpha+\beta}{2}\right)+sin\left(\dfrac{a-\beta+f}{2}\right)-sin\left(\dfrac{f-\left(\alpha+\beta\right)}{2}\right)-sin\left(\dfrac{\alpha+\beta+f}{2}\right)\)

\(=1+sin\left(\dfrac{\pi-2\alpha}{2}\right)+sin\left(\dfrac{\pi-2\beta}{2}\right)-sin\left(\dfrac{2f-\pi}{2}\right)-sin\left(\dfrac{\pi}{2}\right)\)

\(=sin\left(\dfrac{\pi}{2}-\alpha\right)+sin\left(\dfrac{\pi}{2}-\beta\right)+sin\left(\dfrac{\pi}{2}-f\right)\)

\(=cos\alpha+cos\beta+cosf\) (đpcm)

Có sin²a + cos²a = 1

Mà cos a = \(\dfrac{3}{4}\)

=>  sin²a + (\(\dfrac{3}{4}\))² = 1

=> sin²a + \(\dfrac{3^2}{4^2}\) = 1

=> sin²a + \(\dfrac{9}{16}\)= 1

=> sin²a = \(\dfrac{7}{16}\)

=> sin a = \(\dfrac{\sqrt{7}}{4}\)

Có tan a = \(\dfrac{\text{sin a}}{\text{cos a}}\)

Mà \(\left\{{}\begin{matrix}\text{cos a = }\dfrac{3}{4}\\\text{sin a = }\dfrac{\sqrt{7}}{4}\end{matrix}\right.\)

=> tan a = \(\dfrac{\dfrac{\sqrt{7}}{4}}{\dfrac{3}{4}}\) = \(\dfrac{\sqrt{7}}{4}\): \(\dfrac{3}{4}\) = ​\(\dfrac{\sqrt{7}}{4}\).\(\dfrac{4}{3}\) =​\(\dfrac{\sqrt{7}}{3}\)