Lösung 4.4:8c
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using the Pythagorean identity. This means that the equation's left-hand side can be written as | using the Pythagorean identity. This means that the equation's left-hand side can be written as | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\frac{1}{\cos ^{2}x} = \frac{\cos^2\!x + \sin^2\!x}{\cos^2\!x} = 1 + \frac{\sin^2\!x}{\cos^2\!x} = 1+\tan^2\!x</math>}} |
and the expression is then completely expressed in terms of tan x, | and the expression is then completely expressed in terms of tan x, | ||
| - | {{ | + | {{Abgesetzte Formel||<math>1 + \tan^2\!x = 1 - \tan x\,\textrm{.}</math>}} |
If we substitute <math>t=\tan x</math>, we see that we have a quadratic equation in ''t'', which, after simplifying, becomes <math>t^2+t=0</math> and has roots <math>t=0</math> and <math>t=-1</math>. There are therefore two possible values for | If we substitute <math>t=\tan x</math>, we see that we have a quadratic equation in ''t'', which, after simplifying, becomes <math>t^2+t=0</math> and has roots <math>t=0</math> and <math>t=-1</math>. There are therefore two possible values for | ||
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The complete solution of the equation is | The complete solution of the equation is | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\left\{\begin{align} |
x &= n\pi\,,\\[5pt] | x &= n\pi\,,\\[5pt] | ||
x &= \frac{3\pi}{4}+n\pi\,, | x &= \frac{3\pi}{4}+n\pi\,, | ||
Version vom 09:01, 22. Okt. 2008
When we have a trigonometric equation which contains a mixture of different trigonometric functions, a useful strategy can be to rewrite the equation so that it is expressed in terms of just one of the functions. Sometimes, it is not easy to find a way to rewrite it, but in the present case a plausible way is to replace the “1” in the numerator of the left-hand side with \displaystyle \sin^2\!x + \cos^2\!x using the Pythagorean identity. This means that the equation's left-hand side can be written as
| \displaystyle \frac{1}{\cos ^{2}x} = \frac{\cos^2\!x + \sin^2\!x}{\cos^2\!x} = 1 + \frac{\sin^2\!x}{\cos^2\!x} = 1+\tan^2\!x |
and the expression is then completely expressed in terms of tan x,
| \displaystyle 1 + \tan^2\!x = 1 - \tan x\,\textrm{.} |
If we substitute \displaystyle t=\tan x, we see that we have a quadratic equation in t, which, after simplifying, becomes \displaystyle t^2+t=0 and has roots \displaystyle t=0 and \displaystyle t=-1. There are therefore two possible values for \displaystyle \tan x, \displaystyle \tan x=0 or \displaystyle \tan x=-1\,. The first equality is satisfied when \displaystyle x=n\pi for all integers n, and the second when \displaystyle x=3\pi/4+n\pi\,.
The complete solution of the equation is
| \displaystyle \left\{\begin{align}
x &= n\pi\,,\\[5pt] x &= \frac{3\pi}{4}+n\pi\,, \end{align}\right. |
where n is an arbitrary integer.
