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Lösung 4.3:9

Aus Online Mathematik Brückenkurs 1

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Using the formula for double angles on sin160 gives

Vorlage:Displayed math

On the right-hand side, we see that the factor cos80 has appeared, and if we use the formula for double angles on the second factor (sin80),

Vorlage:Displayed math

we obtain a further factor cos40. A final application of the formula for double angles on sin40 gives us all three cosine factors,

Vorlage:Displayed math

We have thus succeeded in showing that

Vorlage:Displayed math

which can also be written as

Vorlage:Displayed math

If we draw the unit circle, we see that 160 makes an angle of 20 with the negative x-axis, and therefore the angles 20 and 160 have the same y-coordinate in the unit circle, i.e.

sin20=sin160.

This shows that

cos80cos40cos20=sin1608sin20=81.
Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_4.3:9“