Symmetry and period of sine and cosine

Trigonometry, grade 10 (5 units) Lesson 5 of 21Learning

Symmetry and period of sine and cosine

The period

\displaystyle 2\pi

The numbers $\displaystyle t$ and $\displaystyle t+2\pi k$ ( $\displaystyle k$ an integer) belong to the same point on the unit circle, so they have the same sine and the same cosine:

\displaystyle \sin(t+2\pi k)

\displaystyle =

\displaystyle \sin t

\displaystyle \cos(t+2\pi k)

\displaystyle =

\displaystyle \cos t

A function that repeats its values every time the same number is added to $\displaystyle t$ is a periodic function. The smallest positive such number is its period: the period of $\displaystyle \sin t$ and of $\displaystyle \cos t$ is $\displaystyle 2\pi$ .

So a large number is reduced by full turns:

$\displaystyle {\sin\frac{25\pi}{6}}={\sin\left(\frac{\pi}{6}+4\pi\right)}={\sin\frac{\pi}{6}}={\frac{1}{2}}$ ;
$\displaystyle {\cos\left(-\frac{17\pi}{3}\right)}={\cos\left(\frac{\pi}{3}-6\pi\right)}={\cos\frac{\pi}{3}}={\frac{1}{2}}$ .

Four reflections, four identities

On the board: the point $\displaystyle {A}={P(t)}$ – here $\displaystyle {t}={0.6}$ – and its reflections.

$\displaystyle {B}={P(-t)}$ – the reflection in the $\displaystyle x$ -axis: the same $\displaystyle x$ , the opposite $\displaystyle y$ .
$\displaystyle {C}={P(\pi-t)}$ – the reflection in the $\displaystyle y$ -axis: the same $\displaystyle y$ , the opposite $\displaystyle x$ .
$\displaystyle {D}={P(\pi+t)}$ – the reflection in the origin: both coordinates opposite.
$\displaystyle {E}={P\left(\frac{\pi}{2}-t\right)}$ – the reflection in the line $\displaystyle {y}={x}$ : the coordinates swap.

\displaystyle x^{2}+y^{2}=1

\displaystyle y=x

\displaystyle A=(\cos0.6,\sin0.6)

\displaystyle B=(\cos0.6,-\sin0.6)

\displaystyle C=(-\cos0.6,\sin0.6)

\displaystyle D=(-\cos0.6,-\sin0.6)

\displaystyle E=(\sin0.6,\cos0.6)

These reflections exist for every $\displaystyle t$ , in every quadrant, so the identities they give hold for every $\displaystyle t$ :

	$\displaystyle -t$	$\displaystyle \pi-t$	$\displaystyle \pi+t$	$\displaystyle \frac{\pi}{2}-t$
$\displaystyle \sin$	$\displaystyle -\sin t$	$\displaystyle \sin t$	$\displaystyle -\sin t$	$\displaystyle \cos t$
$\displaystyle \cos$	$\displaystyle \cos t$	$\displaystyle -\cos t$	$\displaystyle -\cos t$	$\displaystyle \sin t$

For example, $\displaystyle {\sin(\pi-t)}={\sin t}$ and $\displaystyle {\cos(-t)}={\cos t}$ .

Example: simplifying an expression

The example below simplifies the expression $\displaystyle \sin(\pi-x)\sin(-x)+\cos(\pi+x)\cos(-x)$ .

By the identities: $\displaystyle {\sin(\pi-x)}={\sin x}$ , $\displaystyle {\sin(-x)}={-\sin x}$ , $\displaystyle {\cos(\pi+x)}={-\cos x}$ , $\displaystyle {\cos(-x)}={\cos x}$ .
The expression is $\displaystyle {\sin x\cdot(-\sin x)+(-\cos x)\cdot\cos x}={-\sin^2x-\cos^2x}$ .
By the basic identity this is $\displaystyle {-(\sin^2x+\cos^2x)}={-1}$ .

The expression equals $\displaystyle -1$ for every $\displaystyle x$ .

Exercise 1

Find the exact value: first reduce by full turns.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Each expression equals one number for every $\displaystyle x$ . Simplify it with the identities.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Write the expression with $\displaystyle \sin x$ or $\displaystyle \cos x$ alone.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

Dan says: “ $\displaystyle {\sin(x+\pi)}={\sin x}$ for every $\displaystyle x$ , because adding $\displaystyle \pi$ is half a turn.”
Is he right? Explain with the unit circle, and check at $\displaystyle {x}={\frac{\pi}{2}}$ .

Formulas: trigonometry, grade 10 (5 units)

Open as a page

The unit circle (radians)

The unit circle
$\displaystyle x^2+y^2=1$
The point of the number $\displaystyle t$ on the unit circle
$\displaystyle P(t)=(\cos t,\ \sin t)$
All the numbers with the same point as $\displaystyle t$ ( $\displaystyle k$ an integer)
$\displaystyle t+2\pi k$
Radians and degrees
$\displaystyle \pi$ $\displaystyle =$ $\displaystyle 180^{\circ}$
The length of an arc of $\displaystyle \alpha$ radians ( $\displaystyle R$ – the radius)
$\displaystyle \ell$ $\displaystyle =$ $\displaystyle \alpha R$
The area of a sector of $\displaystyle \alpha$ radians ( $\displaystyle R$ – the radius)
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}\alpha R^2$

Special values

$\displaystyle \sin\frac{\pi}{6}=\cos\frac{\pi}{3}=\frac{1}{2}$
$\displaystyle \sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$
$\displaystyle \sin\frac{\pi}{3}=\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$
$\displaystyle \tan\frac{\pi}{6}=\frac{\sqrt{3}}{3},\quad\tan\frac{\pi}{4}=1,\quad\tan\frac{\pi}{3}=\sqrt{3}$

Identities

The basic identity
$\displaystyle \sin^2\alpha+\cos^2\alpha$ $\displaystyle =$ $\displaystyle 1$
Tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$ $\displaystyle \cos\alpha\neq0$
$\displaystyle 1+\tan^2\alpha$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^2\alpha}$
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(\pi-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin(\pi+\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(\pi+\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \sin(\alpha+2\pi k)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \tan(\alpha+\pi k)$ $\displaystyle =$ $\displaystyle \tan\alpha$

Graphs of trigonometric functions

A general sine (and a cosine likewise)
$\displaystyle y=A\sin(ax+b)+B$
The period of $\displaystyle \sin(ax+b)$ and of $\displaystyle \cos(ax+b)$
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{2\pi}{|a|}$
The range of $\displaystyle A\sin(ax+b)+B$
$\displaystyle B-|A|\leq y\leq B+|A|$
The period of $\displaystyle \tan(ax+b)$
$\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{\pi}{|a|}$

Trigonometric equations ($k$ an integer)

The solutions of $\displaystyle {\sin x}={\sin\alpha}$
$\displaystyle x=\alpha+2\pi k,\quad x=\pi-\alpha+2\pi k$
The solutions of $\displaystyle {\cos x}={\cos\alpha}$
$\displaystyle x=\pm\alpha+2\pi k$
The solutions of $\displaystyle {\tan x}={\tan\alpha}$
$\displaystyle x=\alpha+\pi k$

Trigonometry in a triangle (degrees)

In a right-angled triangle: $\displaystyle a$ the leg opposite $\displaystyle \alpha$ , $\displaystyle b$ the leg beside it, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha=\frac{a}{c},\quad\cos\alpha=\frac{b}{c},\quad\tan\alpha=\frac{a}{b}$
$\displaystyle \sin(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
The area of a triangle ( $\displaystyle \alpha$ – the angle between the sides $\displaystyle b$ and $\displaystyle c$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}bc\sin\alpha$
The area of a triangle ( $\displaystyle \beta$ and $\displaystyle \gamma$ – the angles at the side $\displaystyle a$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a^2\sin\beta\sin\gamma}{2\sin(\beta+\gamma)}$
The law of sines ( $\displaystyle R$ – the radius of the circle that circumscribes the triangle)
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines ( $\displaystyle \gamma$ – the angle between the sides $\displaystyle a$ and $\displaystyle b$ )
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2-2ab\cos\gamma$