Odpowiedź:
Szczegółowe wyjaśnienie:
a)
[tex]\frac{2*(sin^{2} 35+ cos^{2}35) }{2*tg35*ctg35 }= \frac{(sin^{2} 35+ cos^{2}35) }{tg35*ctg35 }= \frac{1}{1} =1[/tex]
[tex]\frac{sin a}{cos a} =tg a\\\frac{cosa}{sina } = ctg a[/tex]
[tex]\huge\boxed{a) \ 1}\\\\\\\huge\boxed{b) \ 3-\sqrt{5}}\\\\\\\huge\boxed{c) \ \frac{1}{2}}[/tex]
Korzystamy ze wzorów:
[tex]sin^{2}\alpha + cos^{2}\alpha = 1\\\\tg\alpha\cdot ctg\alpha = 1\\\\tg(90^{0}-\alpha) = ctg\alpha\\\\ctg(90^{0}-\alpha) = tg\alpha[/tex]
Oraz własności potęgowania:
[tex]a^{-n} = \frac{1}{a^{n}}[/tex]
[tex]a) \ \frac{2(sin^{2}35^{0}+cos^{2}35^{0})}{2tg35^{0}\cdot ctg35^{0}}=\frac{2\cdot1}{2\cdot1} =\boxed{ 1}[/tex]
[tex]b) \ 2-\sqrt{5}sin^{2}16^{0}+tg43^{0}\cdot tg47^{0}-\sqrt{5}cos^{2}16^{0} =\\\\=2-\sqrt{5}sin^{2}16^{0}-\sqrt{5}cos^{2}16^{0}+tg43^{0} \cdot tg(90^{0}-43^{0}) =\\\\=2-\sqrt{5}(sin^{2}16^{0}+cos^{2}16^{0})+tg43^{0}\cdot ctg43^{0}=\\\\=2-\sqrt{5}\cdot1+1=2-\sqrt{5}+1 =\boxed{3-\sqrt{5}}[/tex]
[tex]c) \ \left(\frac{tg17^{0}}{ctg73^{0}}+\frac{ctg21^{0}}{tg69^{0}}\right)^{-1}=\left(\frac{tg(90^{0}-73^{0})}{ctg73^{0}}+\frac{ctg(90^{0}-69^{0})}{tg69^{0}}\right)^{-1}=\left(\frac{ctg73^{0}}{ctg73^{0}}+\frac{tg69^{0}}{tg69^{0}}\right)^{-1}=\\\\=(1+1)^{-1} = 2^{-1} = \frac{1}{2^{1}} = \boxed{\frac{1}{2}}[/tex]
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Odpowiedź:
Szczegółowe wyjaśnienie:
a)
[tex]\frac{2*(sin^{2} 35+ cos^{2}35) }{2*tg35*ctg35 }= \frac{(sin^{2} 35+ cos^{2}35) }{tg35*ctg35 }= \frac{1}{1} =1[/tex]
[tex]\frac{sin a}{cos a} =tg a\\\frac{cosa}{sina } = ctg a[/tex]
Verified answer
Odpowiedź:
[tex]\huge\boxed{a) \ 1}\\\\\\\huge\boxed{b) \ 3-\sqrt{5}}\\\\\\\huge\boxed{c) \ \frac{1}{2}}[/tex]
Szczegółowe wyjaśnienie:
Funkcje trygonometryczne
Korzystamy ze wzorów:
[tex]sin^{2}\alpha + cos^{2}\alpha = 1\\\\tg\alpha\cdot ctg\alpha = 1\\\\tg(90^{0}-\alpha) = ctg\alpha\\\\ctg(90^{0}-\alpha) = tg\alpha[/tex]
Oraz własności potęgowania:
[tex]a^{-n} = \frac{1}{a^{n}}[/tex]
[tex]a) \ \frac{2(sin^{2}35^{0}+cos^{2}35^{0})}{2tg35^{0}\cdot ctg35^{0}}=\frac{2\cdot1}{2\cdot1} =\boxed{ 1}[/tex]
[tex]b) \ 2-\sqrt{5}sin^{2}16^{0}+tg43^{0}\cdot tg47^{0}-\sqrt{5}cos^{2}16^{0} =\\\\=2-\sqrt{5}sin^{2}16^{0}-\sqrt{5}cos^{2}16^{0}+tg43^{0} \cdot tg(90^{0}-43^{0}) =\\\\=2-\sqrt{5}(sin^{2}16^{0}+cos^{2}16^{0})+tg43^{0}\cdot ctg43^{0}=\\\\=2-\sqrt{5}\cdot1+1=2-\sqrt{5}+1 =\boxed{3-\sqrt{5}}[/tex]
[tex]c) \ \left(\frac{tg17^{0}}{ctg73^{0}}+\frac{ctg21^{0}}{tg69^{0}}\right)^{-1}=\left(\frac{tg(90^{0}-73^{0})}{ctg73^{0}}+\frac{ctg(90^{0}-69^{0})}{tg69^{0}}\right)^{-1}=\left(\frac{ctg73^{0}}{ctg73^{0}}+\frac{tg69^{0}}{tg69^{0}}\right)^{-1}=\\\\=(1+1)^{-1} = 2^{-1} = \frac{1}{2^{1}} = \boxed{\frac{1}{2}}[/tex]