Trig function of arc trig functions and the reverse

  • #1
461
0
I know the sin(arccos(x)) = (1-x^2)^0.5

I was wondering what some of the others are:

cos(arcsin(X))
tan(arcsin(X))
tan(arccos(x))
sin(arctan(x))
cos(arctan(x))

also the reverse:

arcsin(cos(x))
arcsin(tan(X))
arccos(Sin(X))
arccos(tan(X))
arctan(sin(X))
arctan(cos(X))
 

Answers and Replies

  • #2
22,129
3,297
Let me do two:

[tex]\cos(arcsin(x))=\sqrt{1-\sin(arcsin(x))}=\sqrt{1-x^2}[/tex]

and

[tex]\tan(arccos(x))=\sqrt{\frac{1}{\cos^2(arccos(x))}-1}=\sqrt{\frac{1}{x^2}-1}[/tex]

I'll let you find out the other ones...
 
  • #3
1,065
54
are these always possible? I mean take

arccos(tan(x))

for example, the cosine of an angle is always between 0 and 1, and so, the argument to the arccos function should be a number between 0 and 1...but the tangent of an angle can get pretty large...so, I think these is no solution here...same for others.
 
  • #4
LCKurtz
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Let me do two:


[tex]\tan(arccos(x))=\sqrt{\frac{1}{\cos^2(arccos(x))}-1}=\sqrt{\frac{1}{x^2}-1}[/tex]

Not if arccos(x) is in the second quadrant.
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
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As long as your "angles" are in the first quadrant (so you don't have multi-value problems), you can get all of those formulas by constructing an appropriate right triangle.

For example, to get sin(arctan(x)), imagine a right triangle with "opposite side" x and "near side" 1 (so that the tangent of the angle opposite side "x" is x/1= x and the angle is arctan(x)). By the Pythagorean theorem, it will have "hypotenuse" [itex]\sqrt{x^2+ 1}[/itex]. Sine is "opposite side over hypotenuse" so [itex]sin(arctan(x))= \frac{x}{\sqrt{x^2+ 1}}[/itex].
 

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