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Solution 4.3:3d

From Förberedande kurs i matematik 1

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m (Lösning 4.3:3d moved to Solution 4.3:3d: Robot: moved page)
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{{NAVCONTENT_START}}
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The expression for the angle
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<center> [[Image:4_3_3d.gif]] </center>
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<math>{\pi }/{2}\;-v</math>
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{{NAVCONTENT_STOP}}
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differs from
 +
<math>{\pi }/{2}\;</math>
 +
by as much as
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<math>-v\text{ }</math>
 +
differs from
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<math>0</math>. This means that
 +
<math>{\pi }/{2}\;</math>
 +
makes the same angle with the positive
 +
<math>y</math>
 +
-axis as
 +
<math>-v\text{ }</math>
 +
makes with the positive
 +
<math>x</math>
 +
-axis.
 +
 
 +
 
[[Image:4_3_3_d.gif|center]]
[[Image:4_3_3_d.gif|center]]
 +
 +
Angle
 +
<math>v</math>
 +
angle
 +
<math>\pi -v</math>
 +
 +
 +
Therefore, the angle
 +
<math>{\pi }/{2}\;-v</math>
 +
has a
 +
<math>y</math>
 +
-coordinate which is equal to the
 +
<math>x</math>
 +
-coordinate for the angle
 +
<math>v</math>, i.e.
 +
 +
 +
<math>\sin \left( {\pi }/{2}\;-v \right)=\cos v</math>
 +
 +
 +
and from exercise c, we know that
 +
<math>\cos v=\sqrt{1-a^{2}}</math>
 +
 +
 +
 +
<math>\sin \left( \frac{\pi }{2}-v \right)=\sqrt{1-a^{2}}</math>

Revision as of 11:11, 29 September 2008

The expression for the angle 2−v differs from 2 by as much as −v differs from 0. This means that 2 makes the same angle with the positive y -axis as −v makes with the positive x -axis.


Angle v angle −v


Therefore, the angle 2−v has a y -coordinate which is equal to the x -coordinate for the angle v, i.e.


sin2−v=cosv 


and from exercise c, we know that cosv=1−a2 


sin2−v=1−a2 

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