Turing Trig Expression into Algebraic Expression

Question

I have no idea how to turn this into an algebraic expression, please help ASAP!

Please give me tips on how to do this with other trig to algebraic expressions, as I have four problems I am quite literally clueless on!

 

sin (cos^-1√2/2 - sin^-12x)

 

 

The answer key says the answer is:

 

-2x√2 + √2-8x²

 

 

 

Answer 1

The number sqrt(2)/2  comes up in several places .  To make the typing easier I will replace it with .707 and then restore at the end.  I am going to limit my analysis to The standard domain restriction on sin^-1 .  This restricts x to [-.5 , .5] and generates angles in the first and second quadrant.  Further,  I will take cos^-1(.707) = pi/4 (radian measure) and sin(pi/4) = cos(pi/4) = .707  (first quadrant values)

 

With all of this, the expression is  sin( pi/4 -sin^-1(2 x))  This is the sine of a difference, so the trig angle difference formula can be used.  The result is sin(pi/4) cos(sin^-1(2 x)) - cos(pi/4) sin(sin^-1(2 x)) .  This evaluates to

.707 [ cos(sin^-1(2 x) - sin(sin^-1(2x)) ].

 

With the domain restriction mentioned above sin( sin^-1( 2 x)) = 2 x.      cos( sin^-1(2 x) is a bit more complicated, but a standard triangle analysis show that it is sqrt(1 - 4 x²)  So the expression becomes

  .707[ sqrt( 1 - 4 x²) - 2 x ]  restoring sqrt(2)/2  gives

 

   [sqrt(2)/2] [sqrt(1 - 4 x²) - 2 x]      valid for  -.5 < x < .5

 

The graph of this expression is interesting.   It is clear that it is the top half of a closed curve.   Presumable, the rest of the closed curve could be obtained by properly removing the domain restriction.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Answer 2

If it is written correctly, then

sin (cos^-1√2 - sin^-12x)= [use sin(x-y)-sin(x) cos(y)-cos(x) sin(y)]

=sin(cos^-1√2)cos(sin^-12x)-cos(cos^-1√2)sin(sin^-12x) (⇒)

|sin(sin^-1(x)=x and sin(cos^-1(x)=√(1-x²), therefore|

(⇒)=√(1-(√2)²)√(1-(2x)²)-2√(2) x =√ (-1)√(1-4x²)-2√2 x=

=-2√2 x+i√(1-4x2).