Solution 4.4:3c

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Current revision (12:58, 13 October 2008) (edit) (undo)
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If we consider the entire expression
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If we consider the entire expression <math>x + 40^{\circ}</math> as an unknown, we have a basic trigonometric equation and can, with the aid of the unit circle, see that there are two solutions to the equation for <math>0^{\circ}\le x+40^{\circ}\le 360^{\circ}</math> namely <math>x+40^{\circ} = 65^{\circ}</math> and the symmetric solution <math>x + 40^{\circ} = 180^{\circ} - 65^{\circ} = 115^{\circ}\,</math>.
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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>.
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[[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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It is then easy to set up the general solution by adding multiples of <math>360^{\circ}\,</math>,
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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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{{Displayed math||<math>x + 40^{\circ} = 65^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad x + 40^{\circ} = 115^{\circ} + n\cdot 360^{\circ}</math>}}
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<math>n</math>, which gives
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for all integers ''n'', which gives
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<math>x=2\text{5}^{\circ }+n\centerdot 360^{\circ }</math>
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{{Displayed math||<math>x = 25^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad x=75^{\circ} + n\cdot 360^{\circ}\,\textrm{.}</math>}}
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and
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<math>x=7\text{5}^{\circ }+n\centerdot 360^{\circ }</math>
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Current revision

If we consider the entire expression \displaystyle x + 40^{\circ} as an unknown, we have a basic 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+40^{\circ}\le 360^{\circ} namely \displaystyle x+40^{\circ} = 65^{\circ} and the symmetric solution \displaystyle x + 40^{\circ} = 180^{\circ} - 65^{\circ} = 115^{\circ}\,.

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

\displaystyle x + 40^{\circ} = 65^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad x + 40^{\circ} = 115^{\circ} + n\cdot 360^{\circ}

for all integers n, which gives

\displaystyle x = 25^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad x=75^{\circ} + n\cdot 360^{\circ}\,\textrm{.}
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