From Förberedande kurs i matematik 1

Revision as of 12:52, 2 April 2008 (edit) Oskar (Talk | contribs) (Ny sida: {{NAVCONTENT_START}} <center> Bild:4_1_2.gif </center> {{NAVCONTENT_STOP}}) ← Previous diff		Current revision (07:59, 3 October 2008) (edit) (undo) Tek (Talk | contribs) m
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-	{{NAVCONTENT_START}}	+	If we use the mnemonic that one turn is 360° or <math>2\pi</math> radians, we can derive a formula for the transformation from degrees to radians. Because
-	<center> [[Bild:4_1_2.gif]] </center>	+
-	{{NAVCONTENT_STOP}}	+	{{Displayed math||<math>360\cdot 1^{\circ } = 2\pi\ \text{radians}</math>}}
		+
		+	this gives
		+
		+	{{Displayed math||<math>1^{\circ} = \frac{2\pi}{360}\ \text{radians} = \frac{\pi}{180}\ \text{radians.}</math>}}
		+
		+	Now we can start transforming the angles:
		+
		+
		+	{|
		+	||a)&nbsp;&nbsp;
		+	|width="100%"|<math>45^{\circ} = 45\cdot 1^{\circ} = 45\cdot\frac{\pi}{180}\ \text{radians} = \frac{\pi}{4}\ \text{radians,}</math>
		+	|-
		+	|height="10px"|&nbsp;
		+	|-
		+	||b)&nbsp;&nbsp;
		+	|width="100%"|<math>135^{\circ } = 135\cdot 1^{\circ} = 135\cdot\frac{\pi}{180}\ \text{radians} = \frac{3\pi}{4}\ \text{radians,}</math>
		+	|-
		+	|height="10px"|&nbsp;
		+	|-
		+	||c)&nbsp;&nbsp;
		+	|width="100%"|<math>-63^{\circ} = -63\cdot 1^{\circ} = -63\cdot\frac{\pi}{180}\ \text{radians} = -\frac{7\pi}{20}\ \text{radians,}</math>
		+	|-
		+	|height="10px"|&nbsp;
		+	|-
		+	||d)
		+	|width="100%"|<math>270^{\circ} = 270\cdot 1^{\circ} = 270\cdot\frac{\pi}{180}\ \text{radians} = \frac{3\pi}{2}\ \text{radians.}</math>
		+	|}

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Current revision

If we use the mnemonic that one turn is 360° or \displaystyle 2\pi radians, we can derive a formula for the transformation from degrees to radians. Because

\displaystyle 360\cdot 1^{\circ } = 2\pi\ \text{radians}

this gives

\displaystyle 1^{\circ} = \frac{2\pi}{360}\ \text{radians} = \frac{\pi}{180}\ \text{radians.}

Now we can start transforming the angles:

a)	\displaystyle 45^{\circ} = 45\cdot 1^{\circ} = 45\cdot\frac{\pi}{180}\ \text{radians} = \frac{\pi}{4}\ \text{radians,}
b)	\displaystyle 135^{\circ } = 135\cdot 1^{\circ} = 135\cdot\frac{\pi}{180}\ \text{radians} = \frac{3\pi}{4}\ \text{radians,}
c)	\displaystyle -63^{\circ} = -63\cdot 1^{\circ} = -63\cdot\frac{\pi}{180}\ \text{radians} = -\frac{7\pi}{20}\ \text{radians,}
d)	\displaystyle 270^{\circ} = 270\cdot 1^{\circ} = 270\cdot\frac{\pi}{180}\ \text{radians} = \frac{3\pi}{2}\ \text{radians.}