# Thread: Taylor's Theorem with Lagrange's form of the remainder.

1. ## Taylor's Theorem with Lagrange's form of the remainder.

The question goes like this.

Suppose that two real-valued functions C and S are defined and differentiable on the domain $(-\infty,\infty)$ and satisfy the following properties:

$S'(\theta) = C(\theta), C'(\theta) = - S(\theta), S(0) = 0, C(0) = 1.$

(a) Use Taylor's Theorem with Lagrange's form of the remainder to prove that for all $\theta \in R$,

$1 - \frac{\theta^2}{2} \leq C(\theta) \leq 1 - \frac{\theta^2}{2} + \frac{\theta^4}{24}$

(b) Deduce that the function $C(\theta)$ has no zeros in the interval [0,1.4] but at least one zero in the interval [1.4,1.6]. Let $\lambda$ denote the smallest such zero. Prove that $S(\lambda) = 1, C(\theta + 2\lambda) = -C(\theta)$ and $S(\theta + 2\lambda) = -S(\theta)$

2. Originally Posted by pearlyc
The question goes like this.

Suppose that two real-valued functions C and S are defined and differentiable on the domain $(-\infty,\infty)$ and satisfy the following properties:

$S'(\theta) = C(\theta), C'(\theta) = - S(\theta), S(0) = 0, C(0) = 1.$

(a) Use Taylor's Theorem with Lagrange's form of the remainder to prove that for all $\theta \in R$,

$1 - \frac{\theta^2}{2} \leq C(\theta) \leq 1 - \frac{\theta^2}{2} + \frac{\theta^4}{24}$
These functions ought to remind you of functions called c** and s** that you have met before. That gives a motivation for the definition that follows.

Define $f(\theta) = (S(\theta))^2 + (C(\theta))^2$. Its derivative is $f'(\theta) = 2S(\theta)C(\theta) - 2C(\theta)S(\theta) = 0$. So f is constant, and by evaluating it at 0 you see that the constant is 1. Therefore $(S(\theta))^2 + (C(\theta))^2 = 1$. This shows in particular that $|C(\theta)|\leqslant 1$ for all $\theta$.

Now you can apply Taylor's Theorem with Lagrange's form of the remainder (taking the first two terms plus remainder) to see that $C(\theta) = 1 - \frac{\theta^2}2C(\phi)$ for some $\phi$ between 0 and $\theta$, and this must be $\geqslant 1 - \frac{\theta^2}2$. Now do the same, taking four terms of the Taylor series, plus remainder, to get the other inequality.

Originally Posted by pearlyc
(b) Deduce that the function $C(\theta)$ has no zeros in the interval [0,1.4] but at least one zero in the interval [1.4,1.6]. Let $\lambda$ denote the smallest such zero. Prove that $S(\lambda) = 1, ...$
This follows easily from the intermediate value theorem together with the magic formula $(S(\theta))^2 + (C(\theta))^2 = 1$.

Originally Posted by pearlyc
... $C(\theta + 2\lambda) = -C(\theta)$ and $S(\theta + 2\lambda) = -S(\theta).$
I don't see an easy way to get these results. You want to show that $S(\lambda+\theta) = C(\theta)$ and $C(\lambda+\theta) = -S(\theta)$. The only way I can see to do this is to show that $C(\theta)$ is the sum of its Taylor series $1-\frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \ldots\,,$ and then to show that the function $S(\lambda+\theta)$ is the sum of the same Taylor series.

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