Lösung 4.4:8a

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If we use the formula for double angles,
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<center> [[Image:4_4_8a-1(2).gif]] </center>
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<math>\text{sin 2}x=\text{2sin }x\text{ cos }x</math>, and move all the terms over to the left-hand side, the equation becomes
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<center> [[Image:4_4_8a-2(2).gif]] </center>
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<math>2\sin x\cos x-\sqrt{2}\cos x=0.</math>
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Then, we see that we can take a factor cos x out of both terms,
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<math>\cos x\left( 2\sin x-\sqrt{2} \right)=0</math>
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and hence divide up the equation into two cases. The equation is satisfied either if
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<math>\text{cos }x=0\text{ }</math>
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or if
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<math>2\sin x-\sqrt{2}=0</math>.
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<math>\text{cos }x=0\text{ }</math>: this equation has the general solution
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<math>x=\frac{\pi }{2}+n\pi </math>
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(
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<math>n</math>
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an arbitrary integer)
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<math>2\sin x-\sqrt{2}=0</math>: If we collect
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<math>\text{sin }x</math>
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on the left-hand side, we obtain the equation
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<math>\text{sin }x\text{ }={1}/{\sqrt{2}}\;</math>, which has the general solution
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<math>\left\{ \begin{array}{*{35}l}
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x=\frac{\pi }{4}+2n\pi \\
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x=\frac{3\pi }{4}+2n\pi \\
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\end{array} \right.</math>
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(
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<math>n</math>
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an arbitrary integer)
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The complete solution of the equation is
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<math>\left\{ \begin{array}{*{35}l}
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x=\frac{\pi }{4}+2n\pi \\
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x=\frac{\pi }{2}+n\pi \\
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x=\frac{3\pi }{4}+2n\pi \\
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\end{array} \right.</math>
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(
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<math>n</math>
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an arbitrary integer).

Version vom 13:31, 1. Okt. 2008

If we use the formula for double angles, \displaystyle \text{sin 2}x=\text{2sin }x\text{ cos }x, and move all the terms over to the left-hand side, the equation becomes


\displaystyle 2\sin x\cos x-\sqrt{2}\cos x=0.


Then, we see that we can take a factor cos x out of both terms,


\displaystyle \cos x\left( 2\sin x-\sqrt{2} \right)=0


and hence divide up the equation into two cases. The equation is satisfied either if \displaystyle \text{cos }x=0\text{ } or if \displaystyle 2\sin x-\sqrt{2}=0.


\displaystyle \text{cos }x=0\text{ }: this equation has the general solution


\displaystyle x=\frac{\pi }{2}+n\pi ( \displaystyle n an arbitrary integer)


\displaystyle 2\sin x-\sqrt{2}=0: If we collect \displaystyle \text{sin }x on the left-hand side, we obtain the equation \displaystyle \text{sin }x\text{ }={1}/{\sqrt{2}}\;, which has the general solution


\displaystyle \left\{ \begin{array}{*{35}l} x=\frac{\pi }{4}+2n\pi \\ x=\frac{3\pi }{4}+2n\pi \\ \end{array} \right. ( \displaystyle n an arbitrary integer)

The complete solution of the equation is


\displaystyle \left\{ \begin{array}{*{35}l} x=\frac{\pi }{4}+2n\pi \\ x=\frac{\pi }{2}+n\pi \\ x=\frac{3\pi }{4}+2n\pi \\ \end{array} \right. ( \displaystyle n an arbitrary integer).

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