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Solution 4.3:7a

From Förberedande kurs i matematik 1

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m (Lösning 4.3:7a moved to Solution 4.3:7a: Robot: moved page)
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We can write the expression
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<center> [[Image:4_3_7a.gif]] </center>
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<math>\text{sin}\left( x+y \right)</math>
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in terms of
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<math>\text{sin }x</math>,
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<math>\text{cos }x</math>,
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<math>\text{sin }y</math>
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and
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<math>\text{cos }y</math>
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if we use the addition formula for sine,
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<math>\text{sin}\left( x+y \right)=\sin x\centerdot \cos y+\cos x\centerdot \sin y</math>
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In turn, it is possible to express the factors
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<math>\text{cos }x</math>
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and
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<math>\text{cos }y</math>
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in terms of
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<math>\text{sin }x</math>
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and
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<math>\text{sin }y</math>
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by using the Pythagorean identity,
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<math>\begin{align}
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& \cos x=\pm \sqrt{1-\text{sin}^{2}x}=\pm \sqrt{1-\left( {2}/{3}\; \right)^{2}}=\pm \frac{\sqrt{5}}{3} \\
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& \cos y=\pm \sqrt{1-\text{sin}^{2}y}=\pm \sqrt{1-\left( {1}/{3}\; \right)^{2}}=\pm \frac{2\sqrt{2}}{3} \\
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\end{align}</math>
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Because
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<math>x</math>
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and
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<math>y</math>
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are angles in the first quadrant,
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<math>\text{cos }x</math>
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and
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<math>\text{cos }y</math>
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are positive, so we in fact have
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<math>\cos x=\frac{\sqrt{5}}{3}</math>
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and
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<math>\cos y=\frac{2\sqrt{2}}{3}</math>
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Finally, we obtain
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<math>\sin \left( x+y \right)=\frac{2}{3}\centerdot \frac{2\sqrt{2}}{3}+\frac{\sqrt{5}}{3}\centerdot \frac{1}{3}=\frac{4\sqrt{2}+\sqrt{5}}{9}</math>

Revision as of 10:14, 30 September 2008

We can write the expression sinx+y  in terms of sin x, cos x, sin y and cos y if we use the addition formula for sine,


sinx+y=sinxcosy+cosxsiny 


In turn, it is possible to express the factors cos x and cos y in terms of sin x and sin y by using the Pythagorean identity,


cosx=1−sin2x=1−232=35cosy=1−sin2y=1−132=322


Because x and y are angles in the first quadrant, cos x and cos y are positive, so we in fact have


cosx=35  and cosy=322 


Finally, we obtain


sinx+y=32322+3531=942+5 

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