SOLUTION: Please help me solve this equation: arcsin2x+arcsinx=pi/2

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Question 1162957: Please help me solve this equation: arcsin2x+arcsinx=pi/2
Found 3 solutions by solver91311, Edwin McCravy, ikleyn:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!


You will need a couple of tools to solve this one. First is the cosine of the sum of two angles:



Second, you will need the following identities:



and



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Now use Cosine of the Sum:



Now apply the identities:



The rest is just algebra:









I'm going to presume you know how to rationalize your denominators if necessary, and please note that one of these two solutions is extraneous and must be discarded. I'll leave it to you to figure out which one.


John

My calculator said it, I believe it, that settles it

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
arcsin(2x)+arcsin(x)=π/2
Two angles that have sum π/2 are complementary, thus the sine of one is the
cosine of the other and vice-versa.  So a right triangle that contains such
a pair of complementary angles would be:



The angle with the red arc is arcsin(x) and the angle with the blue arc is
arcsin(2x).

By the Pythagorean theorem,



Since the triangle can be placed in any quadrant, we use ±.

Edwin

Answer by ikleyn(52834) About Me  (Show Source):
You can put this solution on YOUR website!
.

The solution by Edwin is very nice except one thing.

Only positive value of   x = sqrt%285%29%2F5  is the solution.

The negative value   x = - sqrt%285%29%2F5  is not.


Do not make this mistake.