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sin 105 - sin 15

Guest Nov 17, 2014

Best Answer 

 #9
avatar+91479 
+40

Thanks Heureka,

I have to think about that one too Chris :)

It is too late now :))

Melody  Nov 18, 2014
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9+0 Answers

 #1
avatar+17711 
+5

√(2) / 2

geno3141  Nov 18, 2014
 #2
avatar+91479 
+10

 

NOW GENO - I know I am a real expert at making a mountain out of an mole hill BUT how did you do it with no working?   

 

$$\\sin 105 - sin 15\\
=sin(180-105)-sin15\\
=sin75-sin15\\
=cos15-sin15\\
=cos(45-30)-sin(45-30)\\
=cos45cos30+sin45sin30-[sin45cos30-cos45sin30]\\
=sin45cos30+sin45sin30-sin45cos30+cos45sin30\\
=sin45sin30+sin45sin30\\
=2sin45sin30\\
=2\times\frac{1}{\sqrt2}\times\frac{1}{2}\\
=2\times\frac{1}{\sqrt2}\times\frac{1}{2}\\
=\frac{1}{\sqrt2}\\
=\frac{\sqrt2}{2}\\$$

Melody  Nov 18, 2014
 #3
avatar+81063 
+5

Yeah....the way geno came up with that answer is a "sin"

 

CPhill  Nov 18, 2014
 #4
avatar+91479 
+5

Hi Chris,

I am looking forward to Geno response   LOL

Maybe there is some 'obvious' method.  But it will make me feel less stupid if you can't 'see' it either,  :))

Melody  Nov 18, 2014
 #5
avatar+81063 
+5

Don't feel too stupid......I don't see the "trick," either......maybe there is some "identity" of which we are unaware....!!!

 

CPhill  Nov 18, 2014
 #6
avatar+18829 
+5

sin 105 - sin 15

$$\sin{( 105\ensurement{^{\circ}} )} - \sin{( 15\ensurement{^{\circ}} ) } \\
=\sin{( 90\ensurement{^{\circ}} + 15\ensurement{^{\circ}} )} - \sin{( 15\ensurement{^{\circ}} ) } \\
= \underbrace{\sin{( 90\ensurement{^{\circ}} ) }}_{=1}\cos{( 15\ensurement{^{\circ}} ) }+\underbrace{\cos{( 90\ensurement{^{\circ}} ) }}_{=0} \sin{( 15 \ensurement{^{\circ}}) }- \sin{( 15\ensurement{^{\circ}} ) }\\
=\cos{(\textcolor[rgb]{1,0,0}{15\ensurement{^{\circ}}} )} - \sin{( \textcolor[rgb]{1,0,0}{15 \ensurement{^{\circ}}}) } \\ \\
\boxed{
=\sqrt{2}\cos{(\textcolor[rgb]{1,0,0}{15\ensurement{^{\circ}}} +45\ensurement{^{\circ}} )}\\
=-\sqrt{2}\sin{(\textcolor[rgb]{1,0,0}{15\ensurement{^{\circ}}} -45 \ensurement{^{\circ}})} \\
=\frac{\sqrt{2}}{2} = 0.70710678119
}$$

heureka  Nov 18, 2014
 #7
avatar+81063 
0

How did you get from

cos(15) - sin(15)    to

√2cos(15 + 45)   and then to

-√2sin(15 - 45)  ????

 

CPhill  Nov 18, 2014
 #8
avatar+18829 
+5

heureka  Nov 18, 2014
 #9
avatar+91479 
+40
Best Answer

Thanks Heureka,

I have to think about that one too Chris :)

It is too late now :))

Melody  Nov 18, 2014

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