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Trigonometric Equations

  1. Apr 3, 2015 #1
    Is there any way I can reduce or simplify expressions like cos(sin-1(x)), sin(cos-1(x)), cos(tan-1(x)), tan(cot-1(x)) etc.? (I refer to the arc functions, i.e. inverses, by the superscript -1, not the reciprocals)
  2. jcsd
  3. Apr 3, 2015 #2


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    Science Advisor

    Hint: sin(sin-1(x)) = x and cos(w) = sin(w+π/2).
  4. Apr 3, 2015 #3
    You can use this relation:
  5. Apr 3, 2015 #4


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    Draw a right triangle with an angle [itex]\theta[/itex], opposite side length of x and hypotenuse length 1. Now from this triangle, by definition,

    [tex]\sin\theta = x[/tex]
    [tex]\theta = \sin^{-1}x[/tex]

    So then what is [itex]\cos\left(\sin^{-1}x\right)[/itex], or more simply, [itex]\cos\theta[/itex] ?
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