Solution 4.2:3c

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We can add and subtract multiples of <math>2\pi</math> to or from the argument of the sine function without changing its value. The angle <math>2\pi</math> corresponds to a whole turn in a unit circle and the sine function returns to the same value every time the angle changes by a complete revolution.
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We can add and subtract multiples of
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For example, if we can subtract sufficiently many <math>2\pi</math>'s from <math>9\pi</math>, we will obtain a more manageable argument which lies between <math>0</math> and <math>2\pi\,</math>,
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<math>\text{2}\pi </math>
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to or from the argument of the sine function without changing its value. The angle
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<math>\text{2}\pi </math>
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corresponds to a whole turn in a unit circle and the sine function returns to the same value every time the angle changes by a complete revolution.
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For example, if we can subtract sufficiently many
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<math>\text{2}\pi </math>
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s from
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<math>\text{9}\pi </math>, we will obtain a more manageable argument which lies between
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<math>0</math>
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and
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<math>\text{2}\pi </math>,
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<math>\text{sin 9}\pi =\text{sin}\left( 9\pi -2\pi -2\pi -2\pi -2\pi \right)=\sin \pi </math>
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The line which makes an angle
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<math>\pi </math>
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with the positive part of the
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<math>x</math>
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-axis is the negative part of the
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<math>x</math>
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-axis
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and it cuts the unit circle at the point
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<math>\left( -1 \right.,\left. 0 \right)</math>, which is why we can see from the
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<math>y</math>
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-coordinate that
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<math>\text{sin 9}\pi =\text{sin }\pi =0</math>.
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{{Displayed math||<math>\sin 9\pi = \sin (9\pi - 2\pi - 2\pi - 2\pi - 2\pi) = \sin \pi\,\textrm{.}</math>}}
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The line which makes an angle <math>\pi</math> with the positive part of the ''x''-axis is the negative part of the ''x''-axis and it cuts the unit circle at the point (-1,0), which is why we can see from the ''y''-coordinate that <math>\sin 9\pi = \sin \pi = 0\,</math>.
[[Image:4_2_3_c.gif|center]]
[[Image:4_2_3_c.gif|center]]

Current revision

We can add and subtract multiples of \displaystyle 2\pi to or from the argument of the sine function without changing its value. The angle \displaystyle 2\pi corresponds to a whole turn in a unit circle and the sine function returns to the same value every time the angle changes by a complete revolution.

For example, if we can subtract sufficiently many \displaystyle 2\pi's from \displaystyle 9\pi, we will obtain a more manageable argument which lies between \displaystyle 0 and \displaystyle 2\pi\,,

\displaystyle \sin 9\pi = \sin (9\pi - 2\pi - 2\pi - 2\pi - 2\pi) = \sin \pi\,\textrm{.}

The line which makes an angle \displaystyle \pi with the positive part of the x-axis is the negative part of the x-axis and it cuts the unit circle at the point (-1,0), which is why we can see from the y-coordinate that \displaystyle \sin 9\pi = \sin \pi = 0\,.

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