Solution 4.1:2

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m (Lösning 4.1:2 moved to Solution 4.1:2: Robot: moved page)
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If we use the mnemonic that one turn is
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<center> [[Image:4_1_2.gif]] </center>
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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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<math>360^{\circ }\centerdot 1^{\circ }=2\pi </math>
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radians
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this gives
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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:
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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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<math>270^{\circ }=270\centerdot 1^{\circ }=270\centerdot \frac{\pi }{180}</math>
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radians
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<math>=\frac{3\pi }{2}</math>
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radians

Revision as of 09:17, 27 September 2008

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


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

this gives


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

Now we can start transforming the angles:

a) \displaystyle 45^{\circ }=45\centerdot 1^{\circ }=45\centerdot \frac{\pi }{180} radians \displaystyle =\frac{\pi }{4} radians

b) \displaystyle 135^{\circ }=135\centerdot 1^{\circ }=135\centerdot \frac{\pi }{180} radians \displaystyle =\frac{3\pi }{4} radians

c) \displaystyle -63^{\circ }=-63\centerdot 1^{\circ }=-63\centerdot \frac{\pi }{180} radians \displaystyle =-\frac{7\pi }{20} radians

d) \displaystyle 270^{\circ }=270\centerdot 1^{\circ }=270\centerdot \frac{\pi }{180} radians \displaystyle =\frac{3\pi }{2} radians

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