Problem to calculate arcsin(5/24*cos(45)) to a real number

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In Sage as in most math software, angles are measured in radians, not degrees.

So if by cos(45) you mean the cosine of 45 degrees, use cos(pi/4) instead.

The next thing is that Sage aims to give exact results for exact input.

sage: a = arcsin(5/24*cos(pi/4))
sage: a
arcsin(5/48*sqrt(2))

To get a floating-point approximation for your computation, two choices.

• First, you can compute exactly and then take a numerical approximation.

  For this, use the numerical_approx method or its shortcut n.

  sage: a.numerical_approx()
  0.147852003702638
  
  sage: a.n()
  0.147852003702638
  

• Second, you can compute in floating-point all the way.

  For this, start by using a floating point approximation of pi.

  sage: b = asin(5/24*cos(pi.n()/4))
  sage: b
  0.147852003702638
  

I want to have the expression as a real number

arcsin(5/24*cos(45)) is a Sage object representing the quantity you want to manipulate; its value is a real number+. What you are seeking is a numerical approximation of this quantity expressed as a decimal representation. Which can be obtained by :

sage: numerical_approx(arcsin(5/24*cos(45)))
0.109661742042478

or

sage: arcsin(5/24*cos(45)).n()
0.109661742042478

Try numerical_approx?...

• Note that there are an infinity of complex values whose $\sin$ is 5/24*cos(45) ; the mathematical function $\arcsin$ is multivalued. The Sage function arcsin implement aparticular choice of this branch cut, returning a value in the interval $[-\frac\pi2\ \frac\pi2]$.