Re: What is (sin t)^2 equal to?

The question: "What is (sin t)^2 equal to? I don't know if it is sin t^2 or sin^2 t."

The usual interpretation of "(sin t)^2" is that it is identical to "sin^2 t", which is the same as "sin t * sin t". But is this the same as "sin(t^2)"? We can check it by doing some figuring, and also by inputting some angles into the sine function of any scientific calculator.

At mathforum.org we can find a series expression for sine (The angle, x, is in radians.):

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ..... and so

sin(x^2) = x^2 - x^6/3! + x^10/5! - x^14/7! + ......

But if one takes the series for sin(x) and squares it to obtain sin^2(x) one gets

sin^2(x) = x^2 - x^4/3 + 2 x^6/45 - x^8/315 + .....

which doesn't appear to be the same as sin(x^2).

Let's put some angles into our scientific calculator:

The information in the table confirms that sin^2(x) is not the same as sin(x^2).

John Link, MadSci Physicist