Cho cos α = \(\frac{3}{4}\), sin α > 0; sin β = \(\frac{3}{5}\), \(\beta \in \left( {\frac{{9\pi }}{2};\,\,5\pi } \right)\). Hãy tính cos 2α, sin 2α, cos 2β, sin 2β, cos (α + β), sin (α – β).

Lời giải

Ta có cos 2α = 2 cos² α – 1 = \(2.{\left( {\frac{3}{4}} \right)^2} - 1 = \frac{1}{8}\).

Ta có sin² α = 1 – cos² a = \(1 - {\left( {\frac{3}{4}} \right)^2}\)= \(\frac{7}{{16}}\).  

Lại do sin α > 0 nên sin α = \(\frac{{\sqrt 7 }}{4}\).

Suy ra sin 2α = 2 sin α cos α = \(2.\frac{{\sqrt 7 }}{4}.\frac{3}{4} = \frac{{3\sqrt 7 }}{8}\).

Ta có cos 2β = 1 – 2 sin² β = \(1 - 2.{\left( {\frac{3}{5}} \right)^2}\) = \(\frac{7}{{25}}\).

Ta có cos² β = 1 – sin² β = \(1 - {\left( {\frac{3}{5}} \right)^2}\)= \(\frac{{16}}{{25}}\).

Lại do \(\beta \in \left( {\frac{{9\pi }}{2};\,\,5\pi } \right)\) nên cos β < 0, do đó \(\cos \beta = - \frac{4}{5}\).

Suy ra sin 2β = 2 sin β cos β = \(2.\frac{3}{5}.\left( { - \frac{4}{5}} \right) = - \frac{{24}}{{25}}\).

Ta có

cos(α + β) = cos α cos β – sin α sin β = \(\frac{3}{4}.\left( { - \frac{4}{5}} \right) - \frac{{\sqrt 7 }}{4}.\frac{3}{5} = \frac{{ - 12 - 3\sqrt 7 }}{{20}}\).

sin(α – β) = sin α cos β – cos α sin β = \(\frac{{\sqrt 7 }}{4}.\left( { - \frac{4}{5}} \right) - \frac{3}{4}.\frac{3}{5} = \frac{{ - 9 - 4\sqrt 7 }}{{20}}\).