Matematik Opgaver 1738 1740 1736 1737


Indholdsfortegnelse

Opg 1738

Bevis for de eksakte værdier for sin(15) og cos(105). Billede:Definition_af_sin45_og_cos45.png

Definition af sin(45)
LaTex: sin(45)=cos(45)=\frac{\sqrt{1^2+1^2}}{2}=\frac{\sqrt{2}}{2}

Billede:Definition_af_sin30_og_cos30.png

Definition af cos(30)
LaTex: cos(30)=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{\sqrt{4}}=\frac{\sqrt{3}}{2}
Sin(15)
LaTex: sin(15)=sin(45-30)=\sin(45)\cdot\cos(30)-\cos(45)\cdot\sin(30)=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\cdot\frac{1}{2}= \frac{\sqrt{2}\cdot\sqrt{3}}{2\cdot 2}-\frac{\sqrt{2}}{4}= \frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}= \frac{\sqrt{6}-\sqrt{2}}{4}
Cos(105)
LaTex: cos(105) = cos(60+45) = \frac{1}{2}\cdot\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}\cdot\frac{\sqrt{2}}{2}=\frac{\sqrt{2}-\sqrt{6}}{4}

Opg 1740

LaTex: sin( 3x)=sin( 2x+x )=sin( 2x) \cdot cos( x )+cos( 2x) \cdot sin( x)

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Opg 1736

Opg 1737