1. ## Third-degree Taylor polynomial

Problem:

My solution: T3(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7

T3(pi/6) = pi/6 - (pi/6)^3/3 + (pi/6)^5/5 - (pi/6)^7/7 (unsimplified)

Given solution:

I just don't understand the methodology. I thought it was a matter of determining the corresponding Taylor series of the function, writing it out to the nth term specified, and then substituting directly. Yet, that doesn't look like what was done in the given solution. Could someone please help me out?

2. Originally Posted by iheartmath
Problem:

My solution: T3(x) = x - (x^3)/3 + (x^5)/5 - (x^7)/7

T3(pi/6) = pi/6 - (pi/6)^3/3 + (pi/6)^5/5 - (pi/6)^7/7 (unsimplified)

Given solution:

I just don't understand the methodology. I thought it was a matter of determining the corresponding Taylor series of the function, writing it out to the nth term specified, and then substituting directly. Yet, that doesn't look like what was done in the given solution. Could someone please help me out?
When it asks for $\displaystyle T_n(x;x_0)$ it's asking for the Taylor polynomial about $\displaystyle x_0$ whose degree is $\displaystyle n$. You went out too far because you misinterpreted the n to be the number of terms.

3. That makes a lot more sense now. Thanks!

4. As far as the second part of the question, you are asked to approximate the value of $\displaystyle \frac{\pi}{6}$. You will have to try to recall some trigonometry to remember that $\displaystyle \tan \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{3}$. Therefore, $\displaystyle \tan^{-1} \left( \frac{\sqrt{3}}{3} \right) = \frac{\pi}{6}$.

That's why you should use $\displaystyle x=\frac{\sqrt{3}}{3}$ in order for $\displaystyle f(x) \approx \frac{\pi}{6}$.