# Recitation 1

### Binomial numbers and binomial theorem

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.

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.

### List of trig formulas

Try to memorize the following basic formulas:

1. $\cos^2 \alpha + \sin^2 \alpha = 1^2$.
2. $\sin (\pi / 2 - \alpha) = \cos x$, $\cos (\pi / 2 - \alpha) = \sin \alpha$ and $\tan \alpha = \sin \alpha / \cos\alpha$.
3. $\sin(\alpha+\beta) = \sin \alpha\cos \beta + \cos \alpha \sin \beta$.
4. $\cos(\alpha+\beta) = \cos \alpha\cos \beta - \sin \alpha \sin \beta$.

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), $\sin\alpha\sin\beta = ?$, $\cos\alpha\cos\beta = ?$.