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sin(arcsin x + arccos x)

 

answer: 1

how???

Guest May 7, 2017
 #1
avatar+87309 
+1

 

Note that   arcsin x   =   some angle ...  and the arccos x   will be the angle that is complementary  to  this angle

 

To see this....suppose  that x = 1/2....then    arcsin (1/2)  = 30°  

 

And  arccos (1/2)     =  60°

 

So    arcsin x  +  arccos x   =   30° +  60°  = 90°

 

And   sin (90°)  =   1

 

 

cool cool cool 

CPhill  May 7, 2017
 #2
avatar+19653 
0

sin(arcsin x + arccos x)


i)
\(\begin{array}{lrcll} & \cos{(\varphi)} &=& x \\ \text{or}& \quad \varphi &=& \arccos{(x)}\\ \end{array} \)

 

ii)
\(\begin{array}{lrclcl} &\sin{(90^\circ-\varphi)} &=& \cos{(\varphi)} &=& x \\ \text{or}& \quad 90^\circ-\varphi && &=& \arcsin{(x)}\\ \end{array} \)

 

iii)

\(\begin{array}{rcll} && \sin\Big(\arcsin(x)+\arccos(x)\Big) \\ &=& \sin(90^\circ-\varphi+\varphi) \\ &=& \sin(90^\circ)\\ &=& 1\\ \end{array}\)

 

laugh

heureka  May 8, 2017
edited by heureka  May 8, 2017

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