The graphs of sine and cosine

Trigonometry, grade 10 (5 units) Lesson 6 of 21Learning

The graphs of sine and cosine

The graph of sine

As the point $\displaystyle P(x)$ goes round the unit circle, its height is $\displaystyle \sin x$ . On the graph, $\displaystyle x$ is on the horizontal axis and the height $\displaystyle \sin x$ on the upright one. The $\displaystyle x$ -axis is marked in multiples of $\displaystyle \pi$ .

\displaystyle f(x)=\sin x

The properties of $\displaystyle {y}={\sin x}$ :

domain: every $\displaystyle x$ ; range: $\displaystyle -1\leq y\leq1$ ;
period $\displaystyle 2\pi$ : the graph repeats itself every $\displaystyle 2\pi$ ;
zeros: $\displaystyle {x}={\pi k}$ ;
maximum $\displaystyle 1$ at $\displaystyle {x}={\frac{\pi}{2}+2\pi k}$ , minimum $\displaystyle -1$ at $\displaystyle {x}={\frac{3\pi}{2}+2\pi k}$ ;
an odd function: $\displaystyle {\sin(-x)}={-\sin x}$ , and the graph is symmetric about the origin.

( $\displaystyle k$ is an integer, as everywhere in the course.)

The graph of cosine

On the board: $\displaystyle {g(x)}={\cos x}$ , and dashed – $\displaystyle {f(x)}={\sin x}$ . The cosine’s graph has the same shape, shifted:

\displaystyle \cos x

\displaystyle =

\displaystyle \sin\left(x+\frac{\pi}{2}\right)

because $\displaystyle {\sin\left(x+\frac{\pi}{2}\right)}={\sin\left(\pi-\left(\frac{\pi}{2}-x\right)\right)}={\sin\left(\frac{\pi}{2}-x\right)}={\cos x}$ . So the graph of $\displaystyle \cos x$ is the graph of $\displaystyle \sin x$ shifted $\displaystyle \frac{\pi}{2}$ to the left.

\displaystyle f(x)=\sin x

\displaystyle g(x)=\cos x

The properties of $\displaystyle {y}={\cos x}$ :

domain: every $\displaystyle x$ ; range: $\displaystyle -1\leq y\leq1$ ; period $\displaystyle 2\pi$ ;
zeros: $\displaystyle {x}={\frac{\pi}{2}+\pi k}$ ;
maximum $\displaystyle 1$ at $\displaystyle {x}={2\pi k}$ , minimum $\displaystyle -1$ at $\displaystyle {x}={\pi+2\pi k}$ ;
an even function: $\displaystyle {\cos(-x)}={\cos x}$ , and the graph is symmetric about the $\displaystyle y$ -axis.

Exercise 1

Mark on the board all the zeros of $\displaystyle {f(x)}={\sin x}$ and all its extremum points in the part shown.

\displaystyle f(x)=\sin x

Exercise 2

Solve with the help of the graphs.

All the solutions, with $\displaystyle k$ .

Exercise 3

Mark where the graphs of $\displaystyle \sin x$ and $\displaystyle \cos x$ meet in the part shown.
At these points $\displaystyle {\sin x}={\cos x}$ – which points of the unit circle are they?

\displaystyle f(x)=\sin x

\displaystyle g(x)=\cos x

Exercise 4

For which values of $\displaystyle x$ between $\displaystyle 0$ and $\displaystyle 2\pi$ is the cosine positive?

Exercise 5

Write where $\displaystyle {y}={\sin x}$ increases and where it decreases between $\displaystyle 0$ and $\displaystyle 2\pi$ .

Formulas: trigonometry, grade 10 (5 units)

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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 $\displaystyle b$ )
$\displaystyle c^2$ $\displaystyle =$ $\displaystyle a^2+b^2-2ab\cos\gamma$