Solution 4.4:3d

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Current revision (13:02, 13 October 2008) (edit) (undo)
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First, we observe from the unit circle that the equation has two solutions for
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First, we observe from the unit circle that the equation has two solutions for <math>0^{\circ}\le 3x\le 360^{\circ}\,</math>,
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<math>0^{\circ }\le \text{3}x\le \text{36}0^{\circ }</math>,
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<math>3x=15^{\circ }</math>
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and
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<math>3x=180^{\circ }-15^{\circ }=165^{\circ }</math>
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{{Displayed math||<math>3x = 15^{\circ}\qquad\text{and}\qquad 3x = 180^{\circ} - 15^{\circ} = 165^{\circ}\,\textrm{.}</math>}}
[[Image:4_4_3_d.gif|center]]
[[Image:4_4_3_d.gif|center]]
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This means that all of the equation's solutions are
This means that all of the equation's solutions are
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{{Displayed math||<math>3x = 15^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad 3x = 165^{\circ} + n\cdot 360^{\circ}\,,</math>}}
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<math>3x=15^{\circ }+n\centerdot 360^{\circ }</math>
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for all integers ''n'', i.e.
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and
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<math>3x=165^{\circ }+n\centerdot 360^{\circ }</math>
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for all integers
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<math>n</math>, i.e.
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<math>x=5^{\circ }+n\centerdot 120^{\circ }</math>
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{{Displayed math||<math>x = 5^{\circ} + n\cdot 120^{\circ}\qquad\text{and}\qquad x = 55^{\circ} + n\cdot 120^{\circ}\,\textrm{.}</math>}}
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and
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<math>x=55^{\circ }+n\centerdot 120^{\circ }</math>
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Current revision

First, we observe from the unit circle that the equation has two solutions for \displaystyle 0^{\circ}\le 3x\le 360^{\circ}\,,

\displaystyle 3x = 15^{\circ}\qquad\text{and}\qquad 3x = 180^{\circ} - 15^{\circ} = 165^{\circ}\,\textrm{.}

This means that all of the equation's solutions are

\displaystyle 3x = 15^{\circ} + n\cdot 360^{\circ}\qquad\text{and}\qquad 3x = 165^{\circ} + n\cdot 360^{\circ}\,,

for all integers n, i.e.

\displaystyle x = 5^{\circ} + n\cdot 120^{\circ}\qquad\text{and}\qquad x = 55^{\circ} + n\cdot 120^{\circ}\,\textrm{.}
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