The trigonometric functions in radians

Calculus measures angles in radians, as the unit circle of the grade 10 trigonometry course does: an angle of $\displaystyle \alpha$ radians is a central angle on an arc $\displaystyle \alpha$ radii long.

• A whole turn is $\displaystyle 2\pi$ radians, so $\displaystyle \pi$ radians are $\displaystyle 180^{\circ}$.
• $\displaystyle {\frac{\pi}{6}}={30^{\circ}}$, $\displaystyle {\frac{\pi}{4}}={45^{\circ}}$, $\displaystyle {\frac{\pi}{3}}={60^{\circ}}$, $\displaystyle {\frac{\pi}{2}}={90^{\circ}}$, $\displaystyle {\frac{3\pi}{2}}={270^{\circ}}$.
• Every real number $\displaystyle x$ has a point on the unit circle: turn $\displaystyle x$ radians from the positive $\displaystyle x$-axis (anticlockwise when $\displaystyle x>0$). Its coordinates are $\displaystyle (\cos x,\sin x)$.

So $\displaystyle \sin x$ and $\displaystyle \cos x$ are functions of a real number – you can draw them, differentiate them and investigate them like any function. In the tools of the lessons from here on, $\displaystyle x$ is in radians, and the $\displaystyle x$-axis of the boards is numbered in multiples of $\displaystyle \pi$.

The board below shows the graphs of $\displaystyle \sin x$, of $\displaystyle \cos x$ and of $\displaystyle \tan x$ (dashed), from $\displaystyle -2\pi$ to $\displaystyle 2\pi$:

• The domain: $\displaystyle \sin x$ and $\displaystyle \cos x$ are defined for every $\displaystyle x$; $\displaystyle {\tan x}={\frac{\sin x}{\cos x}}$ is not defined where $\displaystyle {\cos x}={0}$, that is where $\displaystyle {x}={\frac{\pi}{2}+\pi k}$ – and there it has vertical asymptotes.
• The values: $\displaystyle -1\leq\sin x\leq1$ and $\displaystyle -1\leq\cos x\leq1$; $\displaystyle \tan x$ takes every value.
• The period: $\displaystyle {\sin(x+2\pi)}={\sin x}$ and $\displaystyle {\cos(x+2\pi)}={\cos x}$ – the period is $\displaystyle 2\pi$; the period of $\displaystyle \tan x$ is $\displaystyle \pi$.
• The zeros: $\displaystyle {\sin x}={0}$ where $\displaystyle {x}={\pi k}$; $\displaystyle {\cos x}={0}$ where $\displaystyle {x}={\frac{\pi}{2}+\pi k}$ ($\displaystyle k$ a whole number).
• Even or odd: $\displaystyle {\cos(-x)}={\cos x}$ – even; $\displaystyle {\sin(-x)}={-\sin x}$ and $\displaystyle {\tan(-x)}={-\tan x}$ – odd.
• The graph of $\displaystyle \cos x$ is the graph of $\displaystyle \sin x$ shifted $\displaystyle \frac{\pi}{2}$ to the left: $\displaystyle {\cos x}={\sin\left(x+\frac{\pi}{2}\right)}$.

A periodic function repeats itself, so it is enough to investigate it on an interval of one period and draw conclusions for the whole line. A period of $\displaystyle f$ is a number $\displaystyle T>0$ for which $\displaystyle {f(x+T)}={f(x)}$ for every $\displaystyle x$.

• If $\displaystyle T$ is a period, so are $\displaystyle 2T$, $\displaystyle 3T$ … “The period” means the smallest period.
• For example, $\displaystyle 2\pi$ is a period of $\displaystyle \cos2x$, but the smallest period is $\displaystyle \pi$: $\displaystyle {\cos2(x+\pi)}={\cos(2x+2\pi)}={\cos2x}$.

The example below finds the period of $\displaystyle {f(x)}={\sin^4x-\cos^4x}$ with identities:

• A difference of squares: $\displaystyle {\sin^4x-\cos^4x}={(\sin^2x-\cos^2x)(\sin^2x+\cos^2x)}$.
• $\displaystyle {\sin^2x+\cos^2x}={1}$, so $\displaystyle {f(x)}={\sin^2x-\cos^2x}={-(\cos^2x-\sin^2x)}={-\cos2x}$.
• The period of $\displaystyle -\cos2x$ is $\displaystyle \pi$. $\displaystyle 2\pi$ is a period of $\displaystyle f$ too, but not the smallest.

The four parameters of $\displaystyle {y}={a\sin(bx+c)+d}$ act on the graph of $\displaystyle \sin x$ as on any function, and each changes one property:

• $\displaystyle a$ – a vertical stretch: the graph goes up and down between $\displaystyle -|a|$ and $\displaystyle |a|$. The number $\displaystyle |a|$ is called the amplitude.
• $\displaystyle b$ – a horizontal squeeze by $\displaystyle b$: the period is $\displaystyle \frac{2\pi}{|b|}$.
• $\displaystyle c$ – a horizontal shift: $\displaystyle {bx+c}={0}$ where $\displaystyle {x}={-\frac{c}{b}}$, and a new period starts there.
• $\displaystyle d$ – a vertical shift: the graph goes up and down around the line $\displaystyle {y}={d}$.

For example $\displaystyle {y}={2\sin2x+1}$, on the board beside $\displaystyle \sin x$ (dashed):

• The amplitude is 2, and the graph moves around $\displaystyle {y}={1}$: the values go from $\displaystyle -1$ to 3.
• $\displaystyle {b}={2}$: the period is $\displaystyle {\frac{2\pi}{2}}={\pi}$ – two periods in every period of $\displaystyle \sin x$.

Going from degrees to radians is a horizontal stretch too: the sine of an angle of $\displaystyle x$ degrees is $\displaystyle \sin\frac{\pi x}{180}$, the graph of $\displaystyle \sin x$ stretched horizontally by $\displaystyle \frac{180}{\pi}\approx57.3$.

The board below shows $\displaystyle \sin x$ (dashed), $\displaystyle \sin\left(x+\frac{\pi}{3}\right)$ – shifted $\displaystyle \frac{\pi}{3}$ to the left – and $\displaystyle \sin\left(x-\frac{\pi}{6}\right)$ – shifted $\displaystyle \frac{\pi}{6}$ to the right:

• The graph of $\displaystyle \sin(x+2\pi)$ is exactly the graph of $\displaystyle \sin x$: a shift by a whole period changes nothing.
• So the same graph comes from several values of $\displaystyle c$: $\displaystyle \sin\left(x+\frac{\pi}{3}\right)$ and $\displaystyle {\sin\left(x+\frac{\pi}{3}-2\pi\right)}={\sin\left(x-\frac{5\pi}{3}\right)}$ are the same function.

Work out without a calculator.

The board shows the graph of a function of the form $\displaystyle {y}={a\sin bx}$.
Read the amplitude and the period from the graph, and write the function on the board.

The function is $\displaystyle {f(x)}={\sin^2x}$.

Write $\displaystyle f(x)$ with $\displaystyle \cos2x$, by the identity $\displaystyle {\cos2x}={1-2\sin^2x}$.

What is the period of $\displaystyle f$?
What values does $\displaystyle f$ take?

The function is $\displaystyle {f(x)}={2\sin x-1}$.

Find all the zeros of $\displaystyle f$: solve the equation $\displaystyle {f(x)}={0}$ (the general solution, with $\displaystyle k$ a whole number).

Mark on the graph the zeros in the interval $\displaystyle -\pi\leq x\leq2\pi$.

Mark on the graph of $\displaystyle {f(x)}={\cos2x+1}$ all the extremum points on the board.

What is the period of $\displaystyle f$, and what values does it take?
Explain by the parameters $\displaystyle a$, $\displaystyle b$ and $\displaystyle d$.