Solution 4.1:2

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Current revision (07:59, 3 October 2008) (edit) (undo)
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If we use the mnemonic that one turn is
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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
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<math>360^{\circ }</math>
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or
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<math>\text{2}\pi </math>
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radians, we can derive a formula for the transformation from degrees to radians. Because
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{{Displayed math||<math>360\cdot 1^{\circ } = 2\pi\ \text{radians}</math>}}
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<math>360^{\circ }\centerdot 1^{\circ }=2\pi </math>
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radians
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this gives
this gives
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{{Displayed math||<math>1^{\circ} = \frac{2\pi}{360}\ \text{radians} = \frac{\pi}{180}\ \text{radians.}</math>}}
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<math>1^{\circ }=\frac{2\pi }{360}</math>
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radians
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<math>=\frac{\pi }{180}</math>
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radians
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Now we can start transforming the angles:
Now we can start transforming the angles:
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a)
 
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<math>45^{\circ }=45\centerdot 1^{\circ }=45\centerdot \frac{\pi }{180}</math>
 
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radians
 
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<math>=\frac{\pi }{4}</math>
 
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radians
 
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b)
 
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<math>135^{\circ }=135\centerdot 1^{\circ }=135\centerdot \frac{\pi }{180}</math>
 
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radians
 
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<math>=\frac{3\pi }{4}</math>
 
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radians
 
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c)
 
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<math>-63^{\circ }=-63\centerdot 1^{\circ }=-63\centerdot \frac{\pi }{180}</math>
 
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radians
 
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<math>=-\frac{7\pi }{20}</math>
 
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radians
 
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d)
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{|
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<math>270^{\circ }=270\centerdot 1^{\circ }=270\centerdot \frac{\pi }{180}</math>
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||a)&nbsp;&nbsp;
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radians
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|width="100%"|<math>45^{\circ} = 45\cdot 1^{\circ} = 45\cdot\frac{\pi}{180}\ \text{radians} = \frac{\pi}{4}\ \text{radians,}</math>
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<math>=\frac{3\pi }{2}</math>
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|-
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radians
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|height="10px"|&nbsp;
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|-
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||b)&nbsp;&nbsp;
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|width="100%"|<math>135^{\circ } = 135\cdot 1^{\circ} = 135\cdot\frac{\pi}{180}\ \text{radians} = \frac{3\pi}{4}\ \text{radians,}</math>
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|-
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|height="10px"|&nbsp;
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|-
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||c)&nbsp;&nbsp;
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|width="100%"|<math>-63^{\circ} = -63\cdot 1^{\circ} = -63\cdot\frac{\pi}{180}\ \text{radians} = -\frac{7\pi}{20}\ \text{radians,}</math>
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|-
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|height="10px"|&nbsp;
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|-
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||d)
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|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.}
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