Solution 4.2:3d

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Current revision (07:57, 9 October 2008) (edit) (undo)
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In order to get an angle between
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In order to get an angle between <math>0</math> and <math>\text{2}\pi</math>, we subtract <math>2\pi</math> from <math>{7\pi }/{2}\,</math>, which also leaves the cosine value unchanged
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<math>0</math>
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and
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<math>\text{2}\pi </math>, we subtract
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<math>\text{2}\pi </math>
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from
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<math>{7\pi }/{2}\;</math>
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, which also leaves the cosine value unchanged
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<math>\cos \frac{7\pi }{2}=\cos \left( \frac{7\pi }{2}-2\pi \right)=\cos \frac{3\pi }{2}</math>
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When we draw a line which makes an angle
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<math>{3\pi }/{2}\;</math>
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with the positive
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<math>x</math>
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-axis, we get the negative
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<math>y</math>
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-axis and we see that this line cuts the unit circle at the point
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<math>\left( 0 \right.,\left. -1 \right)</math>. The
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<math>x</math>
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-coordinate of the intersection point is thus
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<math>0</math>
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and hence
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<math>\cos {7\pi }/{2}\;=\cos {3\pi }/{2}\;=0</math>
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{{Displayed math||<math>\cos\frac{7\pi}{2} = \cos\Bigl(\frac{7\pi}{2}-2\pi\Bigr) = \cos\frac{3\pi}{2}\,\textrm{.}</math>}}
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When we draw a line which makes an angle <math>3\pi/2</math> with the positive ''x''-axis, we get the negative ''y''-axis and we see that this line cuts the unit circle at the point (0,-1). The ''x''-coordinate of the intersection point is thus
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<math>0</math> and hence <math>\cos (7\pi/2) = \cos (3\pi/2) = 0\,</math>.
[[Image:4_2_3_d.gif|center]]
[[Image:4_2_3_d.gif|center]]

Current revision

In order to get an angle between \displaystyle 0 and \displaystyle \text{2}\pi, we subtract \displaystyle 2\pi from \displaystyle {7\pi }/{2}\,, which also leaves the cosine value unchanged

\displaystyle \cos\frac{7\pi}{2} = \cos\Bigl(\frac{7\pi}{2}-2\pi\Bigr) = \cos\frac{3\pi}{2}\,\textrm{.}

When we draw a line which makes an angle \displaystyle 3\pi/2 with the positive x-axis, we get the negative y-axis and we see that this line cuts the unit circle at the point (0,-1). The x-coordinate of the intersection point is thus \displaystyle 0 and hence \displaystyle \cos (7\pi/2) = \cos (3\pi/2) = 0\,.

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