Solution 4.3:3c

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With the help of the Pythagorean identity, we can express
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<center> [[Image:4_3_3c.gif]] </center>
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<math>\cos v</math>
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in terms of
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<math>\text{sin }v</math>,
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<math>\cos ^{2}v+\sin ^{2}v=1</math>
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In addition, we know that the angle
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<math>v</math>
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lies between
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<math>-{\pi }/{2}\;</math>
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and
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<math>{\pi }/{2}\;</math>, i.e. either in the first or fourth quadrant, where angles always have a positive
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<math>x</math>
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-coordinate (cosine value); thus, we can conclude that
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<math>\cos v=\sqrt{1-\text{sin}^{2}\text{ }v}=\sqrt{1-a^{2}}</math>

Revision as of 10:58, 29 September 2008

With the help of the Pythagorean identity, we can express \displaystyle \cos v in terms of \displaystyle \text{sin }v,


\displaystyle \cos ^{2}v+\sin ^{2}v=1


In addition, we know that the angle \displaystyle v lies between \displaystyle -{\pi }/{2}\; and \displaystyle {\pi }/{2}\;, i.e. either in the first or fourth quadrant, where angles always have a positive \displaystyle x -coordinate (cosine value); thus, we can conclude that


\displaystyle \cos v=\sqrt{1-\text{sin}^{2}\text{ }v}=\sqrt{1-a^{2}}

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