Oblicz cos , jeśli:A) sin^2 + cos + 1 = 0B) 3 * sin^2 + 7 * cos + 3 = 0C) 3 * ctg = 2 * sin D).... Question from @BoxPL

BoxPL @BoxPL

October 2018 1 15 Report

Oblicz cos $\alpha$ , jeśli:

A) sin^2 $\alpha$ + cos $\alpha$ + 1 = 0

B) 3 * sin^2 $\alpha$ + 7 * cos $\alpha$ + 3 = 0

C) 3 * ctg $\alpha$ = 2 * sin $\alpha$

D) $\frac{1}{2}$ ( tg $\alpha$ + ctg $\alpha$ ) = $\frac{1}{sin \alpha}$

Paulinka17ck

$A. sin^{2}\alpha + cos\alpha+1=0\\ 1-cos^{2}\alpha + cos\alpha+1=0 \\ -cos^{2}\alpha + cos\alpha +2=0\\ delta = 1+8=9 \\ \sqrt {delta}= 3 \\ cos\alpha =\frac{-1-3}{-2}=2 spr \\ cos\alpha = \frac{-1+3}{-2}=-1$

$B. 3sin^{2}\alpha +7cos\alpha +3=0 \\ 3(1-cos^{2}\alpha) +7cos\alpha +3=0\\ 3- 3cos^{2}\alpha +7cos\alpha +3=0\\ -3cos^{2}\alpha +7cos\alpha +6=0\\ delta = 49+72=121\\ \sqrt{delta}= 11\\ cos\alpha = \frac{-7-11}{6}= 3 spr \\ cos\alpha = \frac{-7+11}{6}=\frac{2}{3}$

$C. 3ctg\alpha=2sin\alpha \\ 3 \frac{cos\alpha}{sin\alpha}=2sin\alpha\\ 3cos\alpha = 2sin^{2}\alpha\\ 3cos\alpha=2(1-cos^{2}\alpha})\\ 3cos\alpha= 2-2cos^{2}\alpha\\ 2cos^{2}\alpha + 3cos\alpha -2=0\\ delta = 9+16=25\\ \sqrt{delta} = 5 \\ cos\alpha= \frac{-3-5}{4}=-2 spr \\ cos\alpha=\frac{-3+5}{4}=\frac{1}{2}$

$D \frac{1}{2}(tg\alpha+ctg\alpha)= \frac{1}{sin\alpha}\\ \frac{1}{2}( \frac{sin\alpha}{cos\alpha} + \frac{cos\alpha}{sin\alpha})= \frac{1}{sin\alpha}\\ \frac{1}{2}*\frac{sin^{2}\alpha + cos^{2}\alpha}{cos\alpha*sin\alpha}= \frac{1}{sin\alpha}\\ \frac{1}{2cos\alpha}=1\\ 1=2cos\alpha \\ cos\alpha = \frac{1}{2}$