>> [music] How can anyone compute sine? I mean nowadays, people use computers or
caluculators. But by what mysterious force are these
computational devices able to perform these computations?
How does the machine know what sine of one is equal to?
So, that's a good question. How do we compute sine of 1 radian?
One way is to use a little bit of calculus.
So, what does calculus tell us? So let's let f of equals sine x, and the
derivative of x is cosine x, the value of f at 0 is 0, sine 00, and the value of the
derivative at 0 is 1, because cosine 0 is 1.
Now, I can use the derivative to approximate the functions value.
The functions value near zero is the functions value at zero, plus how much I
would be the input by, times the ratio of output change to input change, the
derivative at zero. Well given this, in our specific case the
value of function 0 and derivative at 0 is 1, and that means the functions value at
h, for h very small, is about h. That means that sine of a very small angle
is very close to that same value. Let's actually use some numbers.
So very concretely, sine of say 1 32nd is really close to 1 32nd.
Now, in addition to this, we've also got a double angle formula.
The double angle formula for sine says that sine of 2x is 2 times sine x times
cosine x. You can derive this formula from the angle
sum formula for sine. Sine of x plus x gives you this.
Alright. Another way to rewrite that a double angle
formula is to write it entirely in terms of sine.
Alright. Cosine is the square root of 1 minus sine
squared x. Well, as long as the cosine is positive,
this is true. So, if I'm working with a value of x,
which cos sins positive, then sin of 2 x is 2 times sin x times the square root of
1 minus sin squared x. We can put these pieces together.
Well, here's what we do. I know that sin of 1 32nd is very close to
1 32nd because 1 32nd is close to 0. And, sin of a number close to zero is
about that. That same number.
Then I can use this double-angle formula entirely in terms of sign to get an
approximate value for sign of 1 16th by using my approximate value for sign of 1
32nd. Then, I can use this double-angle formula
again to get an approximate value for sign of 1 18th, and again to get an approximate
value for sign of 1 4th, and again to get an approximate value for sign of 1 half,
and then again to get an approximate value of sign of 1.
Which is really quite close to the actual value of sine of 1.
So, what happened? Calculus told us that sine of a small
number, is close to that small number. The double angle formula let us transport
that bit of information around zero to points much further away.
All the way to approximate sine of 1.