Question

выразите cos(70+a) через b если b=sin(40+a)

Answer 1

$b=sin (40textdegree+alpha)$

$cos (70textdegree+alpha)=cos Big(30textdegree+(40textdegree+alpha)Big)=\\=cos 30textdegreecdot cos(40textdegree+alpha) -sin30textdegreecdot sin(40textdegree+alpha)=\\=dfrac{sqrt3}2cdot cos(40textdegree+alpha) -dfrac 12cdot sin(40textdegree+alpha)=\\=pmdfrac{sqrt3}2cdot sqrt{1-sin^2(40textdegree+alpha)} -dfrac 12cdot sin(40textdegree+alpha)=\\=pmdfrac{sqrt3}2cdot sqrt{1-b^2} -dfrac 12cdot b$

$boldsymbol{cos (70textdegree+alpha)=dfrac 12bigg(pmsqrt {3Big(1-b^2Big)}-bbigg)}$

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Использованы формулы

$cos (alpha +beta )=cos alpha cdot cos beta -sin alpha cdot sin beta \\cos ^2 alpha +sin^2=1~~~Rightarrow~~~cosalpha =pmsqrt{1-sin^2alpha }$