jaka to wartośc sinusa
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cienkibolek
- Użytkownik
- Posty: 52
- Rejestracja: 17 mar 2009, o 21:22
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jaka to wartośc sinusa
\(\displaystyle{ sin33=sin(18+15)=sin18cos15+cos18sin15\\
\\
sin15=sin(45-30)=sin45cos30-cos45sin30= \frac{ \sqrt{2} }{2} \cdot \frac{ \sqrt{3} }{2}-\frac{ \sqrt{2} }{2} \cdot \frac{1}{2} = \frac{ \sqrt{6}- \sqrt{2} }{4} \\
\\
cos15=cos(45-30)=cos45cos30+sin45sin30= \frac{ \sqrt{2} }{2} \cdot \frac{ \sqrt{3} }{2}+\frac{ \sqrt{2} }{2} \cdot \frac{1}{2} = \frac{ \sqrt{6}+ \sqrt{2} }{4} \\
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sin18= \frac{ \sqrt{5}-1 }{4} \\
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sin^{2}18+cos^{2}18=1\\
\\
cos18= \sqrt{1-sin^{2}18} = \sqrt{1- \left( \frac{ \sqrt{5}-1 }{4} \right) ^{2}}= \frac{ \sqrt{5+ \sqrt{5} } }{2 \sqrt{2} } \\
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sin33=sin(18+15)=sin18cos15+cos18sin15\\
\\
sin33=\frac{ \sqrt{5}-1 }{4} \cdot \frac{ \sqrt{6}+ \sqrt{2} }{4}+\frac{ \sqrt{5+ \sqrt{5} } }{2 \sqrt{2} } \cdot\frac{ \sqrt{6}- \sqrt{2} }{4}}\)
co do \(\displaystyle{ sin18}\) to 80546.htm
\\
sin15=sin(45-30)=sin45cos30-cos45sin30= \frac{ \sqrt{2} }{2} \cdot \frac{ \sqrt{3} }{2}-\frac{ \sqrt{2} }{2} \cdot \frac{1}{2} = \frac{ \sqrt{6}- \sqrt{2} }{4} \\
\\
cos15=cos(45-30)=cos45cos30+sin45sin30= \frac{ \sqrt{2} }{2} \cdot \frac{ \sqrt{3} }{2}+\frac{ \sqrt{2} }{2} \cdot \frac{1}{2} = \frac{ \sqrt{6}+ \sqrt{2} }{4} \\
\\
sin18= \frac{ \sqrt{5}-1 }{4} \\
\\
sin^{2}18+cos^{2}18=1\\
\\
cos18= \sqrt{1-sin^{2}18} = \sqrt{1- \left( \frac{ \sqrt{5}-1 }{4} \right) ^{2}}= \frac{ \sqrt{5+ \sqrt{5} } }{2 \sqrt{2} } \\
\\
sin33=sin(18+15)=sin18cos15+cos18sin15\\
\\
sin33=\frac{ \sqrt{5}-1 }{4} \cdot \frac{ \sqrt{6}+ \sqrt{2} }{4}+\frac{ \sqrt{5+ \sqrt{5} } }{2 \sqrt{2} } \cdot\frac{ \sqrt{6}- \sqrt{2} }{4}}\)
co do \(\displaystyle{ sin18}\) to 80546.htm
