Binomial numbers and binomial theorem
![](https://blog.zilin.one/wp-content/uploads/2019/01/pascal_triangle.png)
Definition The kth number in the nth row of the Pascal triangle is denoted by {n \choose k}. Careful! The top row is the 0th row, the leftmost numbers are the 0th numbers in the rows.
Theorem [Binomial Theorem] (a+b)^n = {n \choose 0} a^nb^0 + {n \choose 1} a^{n-1}b^1 + {n\choose 2} a^{n-2}b^2 + \dots + {n \choose n} a^0b^n.
Example Expand (1+x)^n and (1-x)^n.
Definition Factorial of n is denoted by n! := 1 x 2 x … x n. Convention: 0! = 1. Fact {n\choose k} = \frac{n!}{k!(n-1)!}.
Further reading Pascal’s pyramid and trinomial expansion.
One-to-one, onto, and bounded functions
Definition (one-to-one) A function f\colon A\to B is one-to-one if for every x_1, x_2 \in A, x_1 \neq x_2 implies f(x_1) \neq f(x_2).
Definition (onto) A function f\colon A\to B is onto if for every y\in B there exists x\in A such that f(x) = y.
Definition (bounded) A function f\colon A\to B is bounded if there exists M \in \mathbb{R} such that for all x\in A, -M \le f(x)\le M.
Examples Function f\colon \mathbb{R} \to \mathbb{R} defined by f(x) = x^3 is both one-to-one and onto, but it is not bounded. However f(x) = \sin x is bounded.
Riddle Find a one-to-one function f\colon \mathbb{R}\to \{\text{irrational numbers}\}.
Graphs of trig function
Graphs of sine, cosine and tangent functions.
![](https://blog.zilin.one/wp-content/uploads/2019/01/trig-graphs.png)
Periodicity of sine and cosine is 2\pi, and the periodicity of tangent is \pi. Sine and cosine functions are bounded, whereas tangent is not bounded.
![](https://blog.zilin.one/wp-content/uploads/2019/01/trig-dance.png)
List of trig formulas
Try to memorize the following basic formulas:
- \cos^2 \alpha + \sin^2 \alpha = 1^2.
- \sin (\pi / 2 - \alpha) = \cos x, \cos (\pi / 2 - \alpha) = \sin \alpha and \tan \alpha = \sin \alpha / \cos\alpha.
- \sin(\alpha+\beta) = \sin \alpha\cos \beta + \cos \alpha \sin \beta.
- \cos(\alpha+\beta) = \cos \alpha\cos \beta - \sin \alpha \sin \beta.
Then you can derive the following formulas:
- \sin 2\alpha = ?, \cos 2\alpha = ?, \tan (\alpha+\beta) = ?, \tan 2\alpha = ?.
- \sin \alpha - \sin \beta = ? (hint: let x=(\alpha+\beta)/2, y=(\alpha-\beta)/2), \cos \alpha - \cos \beta = ?.
- \sin\alpha\cos\beta = ? (hint: replace \beta by -\beta in 3 and add it to 3), \sin\alpha\sin\beta = ?, \cos\alpha\cos\beta = ?.