\(\displaystyle{ \tg30^{\circ}+tg40^{\circ}+tg50^{\circ}+tg60^{\circ}=\frac{4cos20^{\circ}}{cos30^{\circ}}}\)
pomoże ktoś? ;]
Sugerowałabym zapoznać się z regulaminem, a szczególnie punktem III.5.
Wykaż równość (wzory redukcyjne i tożsamości).
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Darnok
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Wykaż równość (wzory redukcyjne i tożsamości).
\(\displaystyle{ (\tg30^{\circ}+tg60^{\circ})+(tg40^{\circ}+tg50^{\circ})}\)
\(\displaystyle{ tg a +tg b= \frac{sin(a+b)}{cos a *cos b}}\)
\(\displaystyle{ \frac{sin 90^{\circ}}{cos30^{\circ}*cos60^{\circ}}+\frac{sin 90^{\circ}}{cos40^{\circ}*cos50^{\circ}}}\)
\(\displaystyle{ \frac{{1}}{cos30^{\circ}*cos60^{\circ}}+\frac{1}{cos40^{\circ}*cos50^{\circ}}}\)
\(\displaystyle{ cos 50^{\circ}=cos (90^{\circ}-40^{\circ})=sin40^{\circ}}\)
\(\displaystyle{ \frac{{1}}{cos30^{\circ}*cos60^{\circ}}+\frac{1}{cos40^{\circ}*sin40^{\circ}}}\)
\(\displaystyle{ sin 2\alpha=2sin\alpha*cos\alpha}\)
\(\displaystyle{ sin\alpha*cos\alpha= \frac{sin 2\alpha}{2}}\)
\(\displaystyle{ cos40^{\circ}*sin40^{\circ}=\frac{sin 80^{\circ}}{2}}\)
\(\displaystyle{ cos60^{\circ}=0,5}\)
\(\displaystyle{ \frac{{2}}{cos30^{\circ}}+\frac{2}{sin80^{\circ}}}\)
\(\displaystyle{ sin80^{\circ}=cos10^{\circ}}\)
\(\displaystyle{ 2(\frac{{1}}{cos30^{\circ}}+\frac{1}{cos10^{\circ}})}\)
\(\displaystyle{ 2(\frac{{cos30^{\circ}+cos10^{\circ}}}{cos30^{\circ}*cos10^{\circ}})}\)
\(\displaystyle{ cos30^{\circ}+cos10^{\circ}=2*cos20^{\circ}*cos10^{\circ}}\)
\(\displaystyle{ 2(\frac{{2*cos20^{\circ}*cos10^{\circ}}}{cos30^{\circ}*cos10^{\circ}})}\)
\(\displaystyle{ 4\frac{{cos20^{\circ}*cos10^{\circ}}}{cos30^{\circ}*cos10^{\circ}}}\)
\(\displaystyle{ 4\frac{{cos20^{\circ}}}{cos30^{\circ}}}\)
\(\displaystyle{ C.K.D.}\)
\(\displaystyle{ tg a +tg b= \frac{sin(a+b)}{cos a *cos b}}\)
\(\displaystyle{ \frac{sin 90^{\circ}}{cos30^{\circ}*cos60^{\circ}}+\frac{sin 90^{\circ}}{cos40^{\circ}*cos50^{\circ}}}\)
\(\displaystyle{ \frac{{1}}{cos30^{\circ}*cos60^{\circ}}+\frac{1}{cos40^{\circ}*cos50^{\circ}}}\)
\(\displaystyle{ cos 50^{\circ}=cos (90^{\circ}-40^{\circ})=sin40^{\circ}}\)
\(\displaystyle{ \frac{{1}}{cos30^{\circ}*cos60^{\circ}}+\frac{1}{cos40^{\circ}*sin40^{\circ}}}\)
\(\displaystyle{ sin 2\alpha=2sin\alpha*cos\alpha}\)
\(\displaystyle{ sin\alpha*cos\alpha= \frac{sin 2\alpha}{2}}\)
\(\displaystyle{ cos40^{\circ}*sin40^{\circ}=\frac{sin 80^{\circ}}{2}}\)
\(\displaystyle{ cos60^{\circ}=0,5}\)
\(\displaystyle{ \frac{{2}}{cos30^{\circ}}+\frac{2}{sin80^{\circ}}}\)
\(\displaystyle{ sin80^{\circ}=cos10^{\circ}}\)
\(\displaystyle{ 2(\frac{{1}}{cos30^{\circ}}+\frac{1}{cos10^{\circ}})}\)
\(\displaystyle{ 2(\frac{{cos30^{\circ}+cos10^{\circ}}}{cos30^{\circ}*cos10^{\circ}})}\)
\(\displaystyle{ cos30^{\circ}+cos10^{\circ}=2*cos20^{\circ}*cos10^{\circ}}\)
\(\displaystyle{ 2(\frac{{2*cos20^{\circ}*cos10^{\circ}}}{cos30^{\circ}*cos10^{\circ}})}\)
\(\displaystyle{ 4\frac{{cos20^{\circ}*cos10^{\circ}}}{cos30^{\circ}*cos10^{\circ}}}\)
\(\displaystyle{ 4\frac{{cos20^{\circ}}}{cos30^{\circ}}}\)
\(\displaystyle{ C.K.D.}\)