Solution 4.3:3f

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In this case, it is perhaps simplest to use the addition formula for sine,
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<math>\sin \left( \frac{\pi }{3}+v \right)=\sin \frac{\pi }{3}\centerdot \cos v+\cos \frac{\pi }{3}\centerdot \sin v.</math>
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Since
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<math>\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2},\ \ \cos \frac{\pi }{3}=\frac{1}{2},\ \ \sin v=a</math>, and
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<math>\cos v=\sqrt{1-a^{2}}</math>
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this can be written as
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<math>\sin \left( \frac{\pi }{3}+v \right)=\frac{\sqrt{3}}{2}\sqrt{1-a^{2}}+\frac{1}{2}a.</math>

Revision as of 11:34, 29 September 2008

In this case, it is perhaps simplest to use the addition formula for sine,


\displaystyle \sin \left( \frac{\pi }{3}+v \right)=\sin \frac{\pi }{3}\centerdot \cos v+\cos \frac{\pi }{3}\centerdot \sin v.

Since \displaystyle \sin \frac{\pi }{3}=\frac{\sqrt{3}}{2},\ \ \cos \frac{\pi }{3}=\frac{1}{2},\ \ \sin v=a, and \displaystyle \cos v=\sqrt{1-a^{2}} this can be written as


\displaystyle \sin \left( \frac{\pi }{3}+v \right)=\frac{\sqrt{3}}{2}\sqrt{1-a^{2}}+\frac{1}{2}a.

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