Question

Решите пж !!!!!!!!!!!

Answer 1

$\sin \alpha = -\dfrac{3}{5}\ ,\ \dfrac{3}{2}\pi < \alpha < 2\pi\ ,\ \cos\alpha>0$

По основному тригонометрическому тождеству:

$\cos^2\alpha + \sin^2\alpha = 1\\\\\cos\alpha = \sqrt{1 - \sin^2\alpha} = \sqrt{1-\left(-\dfrac{3}{5}\right)^2} = \sqrt{1 - \dfrac{9}{25}} = \sqrt{\dfrac{16}{25}} = \dfrac{4}{5}$

$\sin\beta = \dfrac{8}{17}\ ,\ 0<\beta<\dfrac{\pi}{2}\ ,\ \cos\beta > 0$

По основному тригонометрическому тождеству:

$\cos\beta = \sqrt{1-\sin^2\beta} = \sqrt{1 - \left(\dfrac{8}{17}\right)^2} = \sqrt{1 - \dfrac{64}{289}} = \sqrt{\dfrac{225}{289}} = \dfrac{15}{17}$

$\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta = \dfrac{4}{5}\cdot \dfrac{15}{17} - \left(-\dfrac{3}{5}\right)\cdot \dfrac{8}{17} = \dfrac{4\cdot 15}{5\cdot 17} + \dfrac{3\cdot 8}{5\cdot 17} =\\\\\\= \dfrac{4\cdot 15 + 3\cdot 8}{5\cdot 17} = \dfrac{60 + 24}{85} = \boxed{\bf{\dfrac{84}{85}}}$

$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta = \dfrac{4\cdot 15}{5\cdot 17} - \dfrac{3\cdot 8}{5\cdot 17} = \dfrac{4\cdot 15 - 3\cdot 8}{5\cdot 17} = \dfrac{60-24}{85} = \\\\= \boxed{\bf{\dfrac{36}{85}}}$