SOLUTION: If x > 0, {{{ ( arcsin ( x/2 ) + arccos ( x/2 ) ) / arctan ( x ) = 3/2 }}}, solve for x.

Question 1176724: If x > 0, , solve for x.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!

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Added after seeing the response from tutor @ikleyn....

She used a nice fact that I overlooked that makes solving this problem relatively easy.

The numerator is the sum of two angles, each less than pi/2 or 90 degrees, in which the sine of one angle is the cosine of the other. But that makes the two angles complementary, so the numerator of the fraction is just pi/2 (radians), or 90 degrees.

Then from there the solution is relatively easy.

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To solve this algebraically, I think you would have to use calculus and the Taylor series expressions for arcsin, arccos, and arctan. And even then solving the problem would probably involve very ugly calculations.

On the other hand, solving the problem using a graphing calculator is easy -- simply graph the expressions on the two sides of the equation and find where they are equal.

Note that the domains of arcsin and arctan are from -pi/2 to pi/2 and the domain of arccos is from 0 to pi; so the domain of the expression shown is from 0 to pi/2.

HOWEVER....

The topic you have chosen for your post is trigonometry basics -- and solving the problem either of those ways is not basics.

If solving the problem involves basic trigonometry, then the answer likely lies with one of the "nice" acute angles. And for "nice" acute angles, with the "x/2" as the argument for the sine and cosine and "x" as the argument for tangent, x=sqrt(3) seems a likely candidate.

And, indeed, with the angles in degrees,

and

ANSWER: x = sqrt(3)

Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
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if x > 0,     solve for x      = .

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                    Solution

Let a = , b = . It means that sin(a) = ; cos(b) = and "a" and "b" are the angles in QI (acute angles). Since sin(a) = cos(b), it means that a + b = . In other words, the numerator in our fraction, which is + , is equal to . Thus our original equation is = . It implies arctan(x) = = . Since arctan(x) = , it implies x = = . ANSWER. x = .

Solved.

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