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Solution 4.3:3d

From Förberedande kurs i matematik 1

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The expression for the angle <math>\pi/2 - v</math> differs from <math>\pi/2</math> by as much as <math>-v</math> differs from <math>0</math>. This means that <math>\pi/2-v</math> makes the same angle with the positive ''y''-axis as <math>-v</math> makes with the positive ''x''-axis.
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<center> [[Bild:4_3_3d.gif]] </center>
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{| align="center"
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| align="center" |[[Image:4_3_3_d-1.gif]]
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| width="20px"|&nbsp;
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| align="center" |[[Image:4_3_3_d-2.gif]]
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|-
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| align="center" |<small>Angle&nbsp;''v''</small>
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| align="center" |<small>Angle&nbsp;π/2&nbsp;-&nbsp;''v''</small>
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|}
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Therefore, the angle <math>\pi/2 - v</math> has a ''y''-coordinate which is equal to the ''x''-coordinate for the angle ''v'', i.e.
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{{Displayed math||<math>\sin\Bigl(\frac{\pi}{2} - v\Bigr) = \cos v</math>}}
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and from exercise c, we know that <math>\cos v = \sqrt{1-a^2}\,</math>,
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{{Displayed math||<math>\sin\Bigl(\frac{\pi}{2}-v\Bigr) = \sqrt{1-a^2}\,\textrm{.}</math>}}

Current revision

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

Image:4_3_3_d-1.gif   Image:4_3_3_d-2.gif
Angle v Angle π/2 - 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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