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Solution 4.2:4b

From Förberedande kurs i matematik 1

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m (Lösning 4.2:4b moved to Solution 4.2:4b: Robot: moved page)
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We start by subtracting
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<center> [[Image:4_2_4b-1(2).gif]] </center>
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<math>2\pi </math>
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{{NAVCONTENT_STOP}}
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from
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{{NAVCONTENT_START}}
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<math>\frac{11\pi }{3}</math>, so that we get an angle between
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<center> [[Image:4_2_4b-2(2).gif]] </center>
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<math>o</math>
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{{NAVCONTENT_STOP}}
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and
 +
<math>2\pi </math>. This doesn't change the cosine value
 +
 
 +
 
 +
<math>\cos \frac{11\pi }{3}=\cos \left( \frac{11\pi }{3}-2\pi \right)=\cos \frac{5\pi }{3}</math>
 +
 
 +
 
 +
Then, by rewriting
 +
<math>\frac{5\pi }{3}</math>
 +
as a sum of
 +
<math>\pi </math>
 +
- and
 +
<math>\frac{\pi }{2}</math>
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-terms
 +
 
 +
 
 +
<math>\frac{5\pi }{3}=\frac{3\pi +\frac{3}{2}\pi +\frac{1}{2}\pi }{3}=\pi +\frac{\pi }{2}+\frac{\pi }{6}</math>
 +
 
 +
we see that
 +
<math>\frac{5\pi }{3}</math>
 +
is an angle in the fourth quadrant which makes an angle
 +
<math>\frac{\pi }{6}</math>
 +
with the negative
 +
<math>y</math>
 +
-axis.
 +
 
 +
 
[[Image:4_2_4b1.gif]]
[[Image:4_2_4b1.gif]]
 +
 +
With the help of a triangle and a little trigonometry, we can determine the coordinates for the point on a unit circle which corresponds to the angle
 +
<math>\frac{5\pi }{3}</math> .
 +
 +
[[Image:4_2_4_b2.gif]]
[[Image:4_2_4_b2.gif]]
 +
 +
The point has coordinates
 +
<math>\left( \frac{1}{2} \right.,\left. -\frac{\sqrt{3}}{2} \right)</math>
 +
and
 +
<math>\cos \frac{11\pi }{3}=\cos \frac{5\pi }{3}=\frac{1}{2}</math>.

Revision as of 13:05, 28 September 2008

We start by subtracting 2 from 311, so that we get an angle between o and 2. This doesn't change the cosine value


cos311=cos311−2=cos35 


Then, by rewriting 35 as a sum of - and 2 -terms


35=33+23+21=+2+6

we see that 35 is an angle in the fourth quadrant which makes an angle 6 with the negative y -axis.


Image:4_2_4b1.gif

With the help of a triangle and a little trigonometry, we can determine the coordinates for the point on a unit circle which corresponds to the angle 35 .


Image:4_2_4_b2.gif

The point has coordinates 21−23  and cos311=cos35=21.

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