Processing Math: Done

From Förberedande kurs i matematik 1

Revision as of 09:58, 9 September 2008 (edit) Tekbot (Talk | contribs) m (Lösning 4.3:8b moved to Solution 4.3:8b: Robot: moved page) ← Previous diff		Revision as of 10:53, 30 September 2008 (edit) (undo) Ian (Talk | contribs) Next diff →
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-	{{NAVCONTENT_START}}	+
-	<center> [[Image:4_3_8b.gif]] </center>	+
-	{{NAVCONTENT_STOP}}	+	Because
		+	<math>\tan v=\frac{\sin v}{\cos v}</math>, the left-hand side can be written using
		+	<math>\cos v</math>
		+	as the common denominator:
		+
		+
		+	<math>\frac{1}{\cos v}-\tan v=\frac{1}{\cos v}-\frac{\sin v}{\cos v}=\frac{\text{1-}\sin v}{\cos v}</math>
		+
		+
		+	Now, we observe that if we multiply top and bottom by with
		+	<math>\text{1}+\sin v</math>, the denominator will contain the denominator of the right-hand side as a factor and, in addition, the numerator can be simplified to give
		+	<math>\text{1}-\sin ^{2}v\text{ }=\cos ^{2}v</math>, using the conjugate rule:
		+
		+
		+	<math>\begin{align}
		+	& \frac{\text{1-}\sin v}{\cos v}=\frac{\text{1-}\sin v}{\cos v}\centerdot \frac{1+\sin v}{1+\sin v}=\frac{1-\sin ^{2}v}{\cos v\left( 1+\sin v \right)} \\
		+	& =\frac{\cos ^{2}v}{\cos v\left( 1+\sin v \right)}. \\
		+	\end{align}</math>
		+
		+
		+	Eliminating
		+	<math>\cos v</math>
		+	then gives the answer:
		+
		+
		+	<math>\frac{\cos ^{2}v}{\cos v\left( 1+\sin v \right)}=\frac{\cos v}{1+\sin v}</math>

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Revision as of 10:53, 30 September 2008

Because tanv=sinvcosv, the left-hand side can be written using cosv as the common denominator:

1cosv−tanv=1cosv−sinvcosv=cosv1-sinv

Now, we observe that if we multiply top and bottom by with 1+sinv, the denominator will contain the denominator of the right-hand side as a factor and, in addition, the numerator can be simplified to give 1−sin2v =cos2v, using the conjugate rule:

cosv1-sinv=cosv1-sinv1+sinv1+sinv=1−sin2vcosv1+sinv=cos2vcosv1+sinv

Eliminating cosv then gives the answer:

cos2vcosv1+sinv=cosv1+sinv