Why the Sine Graph Repeats Itself Forever

Trigonometric functions repeat. Every full turn brings sine and cosine back to the same values, and that repetition makes them the natural language for tides, sound, alternating current and anything else that cycles.

This part covers proving identities, periods, the graphs of all six functions, and using graphs and periodicity for angles of any size.

A trigonometric identity is proved by starting from one side — usually the more complicated one — and transforming it into the other using the reciprocal, quotient and Pythagorean relations.

Tools:

- Reciprocal: $\text{cosec}\theta =\frac{1}{\sin \theta }$, $\sec \theta =\frac{1}{\cos \theta }$, $\cot \theta =\frac{1}{\tan \theta }$
- Quotient: $\tan \theta =\frac{\sin \theta }{\cos \theta }$
- Pythagorean: ${\sin }^2\theta +{\cos }^2\theta =1$, $1+{\tan }^2\theta ={\sec }^2\theta$, $1+{\cot }^2\theta ={\text{cosec}}^2\theta$

Worked example. Prove $\frac{1-\cos \theta }{\sin \theta }=\frac{\sin \theta }{1+\cos \theta }$.

Multiply the left side above and below by $1+\cos \theta$:

$$\frac{(1-\cos \theta )(1+\cos \theta )}{\sin \theta (1+\cos \theta )}=\frac{1-{\cos }^2\theta }{\sin \theta (1+\cos \theta )}=\frac{{\sin }^2\theta }{\sin \theta (1+\cos \theta )}=\frac{\sin \theta }{1+\cos \theta }$$

Numerical check. At $\theta =60°$, the left side is $\frac{0.5}{0.866}\approx 0.577$ and the right side is $\frac{0.866}{1.5}\approx 0.577$.

An everyday example. Checking a bank passbook balance two ways — from the deposits or from the withdrawals — must give the same figure, just as both sides of an identity must agree for every angle.

The substance. Never work on both sides at once or cross-multiply — that assumes the very identity you are trying to prove.

**A function is periodic with period T if $f(x+T)=f(x)$ for every x, with T the smallest such positive number; sine, cosine, secant and cosecant have period $2\pi$, while tangent and cotangent have period $\pi$.

Periods:**

- $\sin x$, $\cos x$, $\sec x$ and $\text{cosec}x$ — period $2\pi$
- $\tan x$ and $\cot x$ — period $\pi$, since $\tan (x+\pi )=\frac{-\sin x}{-\cos x}=\tan x$

Scaling the input. If f has period T, then $f(ax+b)$ has period $\frac{T}{|a|}$.

Worked example.

- $\sin 3x$ has period $\frac{2\pi }{3}$
- $\tan \frac{x}{2}$ has period $\pi ÷\frac{1}{2}=2\pi$
- $\cos (4x+1)$ has period $\frac{2\pi }{4}=\frac{\pi }{2}$

An everyday example. Household electricity in India alternates at 50 hertz, so the voltage follows a sine curve with a period of $\frac{1}{50}=0.02$ seconds.

The substance. Adding a constant shifts a graph but never changes its period — only the coefficient of x does.

**Sine and cosine are smooth waves between $-1$ and $1$, tangent and cotangent are repeating branches separated by vertical asymptotes, and secant and cosecant are U-shaped branches lying outside the band from $-1$ to $1$.

Key features on $[0,2\pi ]$:**

- $y=\sin x$ — starts at 0, peaks at 1 at $\frac{\pi }{2}$, returns to 0 at $\pi$, falls to $-1$ at $\frac{3\pi }{2}$
- $y=\cos x$ — the sine wave shifted left by $\frac{\pi }{2}$, so it starts at 1
- $y=\tan x$ — zero at 0 and $\pi$; asymptotes at $\frac{\pi }{2}$ and $\frac{3\pi }{2}$
- $y=\cot x$ — asymptotes at 0, $\pi$ and $2\pi$; zero at $\frac{\pi }{2}$
- $y=\sec x$ — U-shapes touching $1$ and $-1$; asymptotes where $\cos x=0$
- $y=\text{cosec}x$ — the same pattern built on $\sin x$

Worked example. Plot $y=2\sin x$ at $x=0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi$: the values are $0,2,0,-2,0$. The wave keeps the period $2\pi$ but its height doubles.

An everyday example. The rise and fall of the sea along the Mumbai coast through a day follows a roughly sine-shaped curve.

The substance. **The asymptotes of $\sec x$ fall exactly where $\cos x=0$** — a reciprocal graph shoots off wherever the original graph crosses zero.

**Because trigonometric functions repeat, any angle can be reduced by whole periods to one between 0 and $2\pi$, and the graph then shows every angle at which a function takes a given value.

Reducing large angles:**

- $\sin (2n\pi +x)=\sin x$ and $\cos (2n\pi +x)=\cos x$ for any integer n
- $\tan (n\pi +x)=\tan x$

Worked example. Find $\sin 765°$. Since $765°=2\times 360°+45°$,

$$\sin 765°=\sin 45°=\frac{1}{\sqrt{2}}\approx 0.707$$

Worked example 2. Solve $\sin x=\frac{1}{2}$ for $0\le x\le 2\pi$. The line $y=\frac{1}{2}$ meets the sine curve twice in this interval: $x=\frac{\pi }{6}$ and $x=\pi -\frac{\pi }{6}=\frac{5\pi }{6}$. Adding $2n\pi$ to each gives every solution.

An everyday example. The hour hand of a clock after 14 hours points where it pointed after 2 hours — removing whole turns is exactly what periodicity does.

The substance. Negative angles follow simple rules too — $\sin (-x)=-\sin x$ and $\cos (-x)=\cos x$, so $\cos (-60°)=\frac{1}{2}$.