Solution 4.4:3c

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m (Lösning 4.4:3c moved to Solution 4.4:3c: Robot: moved page)
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If we consider the entire expression
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<center> [[Image:4_4_3c.gif]] </center>
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<math>x+\text{4}0^{\circ }</math>
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as an unknown, we have a fundamental trigonometric equation and can, with the aid of the unit circle, see that there are two solutions to the equation for
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<math>0^{\circ }\le x+\text{4}0^{\circ }\le \text{36}0^{\circ }</math>
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namely
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<math>x+\text{4}0^{\circ }=\text{65}^{\circ }</math>
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and the symmetric solution
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<math>x+\text{4}0^{\circ }=\text{18}0^{\circ }-\text{65}^{\circ }=\text{115}^{\circ }</math>.
[[Image:4_4_3_c.gif|center]]
[[Image:4_4_3_c.gif|center]]
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It is then easy to set up the general solution by adding multiples of
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<math>360^{\circ }</math>,
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<math>x+\text{4}0^{\circ }=\text{65}^{\circ }+n\centerdot 360^{\circ }</math>
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and
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<math>x+\text{4}0^{\circ }=\text{115}^{\circ }+n\centerdot 360^{\circ }</math>
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for all integers
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<math>n</math>, which gives
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<math>x=2\text{5}^{\circ }+n\centerdot 360^{\circ }</math>
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and
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<math>x=7\text{5}^{\circ }+n\centerdot 360^{\circ }</math>

Revision as of 09:50, 1 October 2008

If we consider the entire expression \displaystyle x+\text{4}0^{\circ } as an unknown, we have a fundamental trigonometric equation and can, with the aid of the unit circle, see that there are two solutions to the equation for \displaystyle 0^{\circ }\le x+\text{4}0^{\circ }\le \text{36}0^{\circ } namely \displaystyle x+\text{4}0^{\circ }=\text{65}^{\circ } and the symmetric solution \displaystyle x+\text{4}0^{\circ }=\text{18}0^{\circ }-\text{65}^{\circ }=\text{115}^{\circ }.


It is then easy to set up the general solution by adding multiples of \displaystyle 360^{\circ },


\displaystyle x+\text{4}0^{\circ }=\text{65}^{\circ }+n\centerdot 360^{\circ } and \displaystyle x+\text{4}0^{\circ }=\text{115}^{\circ }+n\centerdot 360^{\circ }


for all integers \displaystyle n, which gives


\displaystyle x=2\text{5}^{\circ }+n\centerdot 360^{\circ } and \displaystyle x=7\text{5}^{\circ }+n\centerdot 360^{\circ }

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