hTRIGONOMETRIC FUNCTION AND ITS GRAPHIC

A. Definition of Trigonometry

The term comes from two words , namely "trigonos" and "metron". “Trigonos” means

triangle and “metron” meaning measure.

Trigonometry means one unit of mathematics that describes the triangle measures

includes the amount of corners, the amount of sides, angles, and trigonometric functions.

Radian

A long time ago, people often used radian as the unit of degree.

In a radiant system, 1 radian is the angle of the magnitude of angle whose arc length

in front of the angle is equal to the radius of the circle.

∠POQ= P̂Qr

Radiant = rr

= 1 radiant

The angle of one full rotation is 360 °.

360 ° = 2 πr

r=¿2π radians,

Then 180 ° = π radians.

Thus we can conclude that xo= x . π

180rad or xπ rad=xπ .

180o

π

B. Trigonometric function

Function that mapped the set of angle xo to the set of real number sin xo is called sinus

function. Denoted as f : xo sin xo.

Or it can be written as f (xo)=sin xo or f ( x )=sin x , x∈radiant .

Function that mapped the set of angle xo to the set of real number cos xo is called

cosines function. Denoted as f : xo cos xo.

Or it can be written as f (xo)=cos xo or f ( x )=cos x , x∈radiant

Function that mapped the set of angle xo to the set of real number tan xo is called

tangent function. Denoted as f : xo tan xo.

Or it can be written as f (xo)=tan xo or f ( x )=tan x , x∈radiant

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C. Deriving Trigonometric Function

Deriving sinus, cosines, and tangent of 45o

Consider a right isosceles triangle ACB, both legs have the same length.

If you know one leg (a), you know both legs and you can use the Pythagorean

Theorem to find the hypotenuse c.

So these are the proportions of the sides of a 45-45-90 triangle

In other words, the ratio of the hypotenuse to either leg is

√2: 1. In terms of trigonometry,

Sin (45o) = sin (A)

= BCAB

= 1

√2

= 12√2

Thus, it has been proven that

2

C =

=

=

=

Sin 45o =

We can also prove the value of cosines 45o.

Cos (45o)= cos (A)

= ACAB

= 1

√2

= 12√2

So, it can be concluded that

Thus tan 45o = sin 45o

cos45o =

12

√2

12

√2 = 1

Deriving sinus, cosines, and tangent of 30o and 60o

From an equilateral triangle and by using Pythagoras theorem, we can get sine, cosine,

tangent, cosec, sec, cot of 30o and 60o.

a. Sinus Function

Deriving use unit circle

We draw the unit circle, the circle of radius one. We use a positive reference direction

counter-clockwise.

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Cos 45o =

Sin 60 o = ½ √ 3

Cos 60 o = ½

Tan 60 o = √ 3

Cosec 60 o = 2/3 √ 3

Sec 60 o = 2

Cot 60 o = 1/3 √ 3

Sin 30 o = ½ √ 3

Cos 30 o = ½ √ 3

Tan 30 o = 1/3 √ 3

Cosec 30 o = 2

Sec 30 o = 2/3 √ 3

Cot 30 o = √ 3

a. Sinus Function

By making radius r = OP = 1, then:

sin θ=PQr

=PQ

PQ=0 when θ=0 , and larger if θ larger until maximum PQ = 1 when θ=90. Then PQ

decrease until PQ=0 when θ=180 o. After that, PQ became negative and minimum

PQ=-1 when θ=270 o, and then increase until PQ=0 when θ=360. After that , the

event will repeat and the next previous happen when θ=720 o. The event repeat again

and so on. A cycle event is called periodic. Then we conclude :

Sin 0o = 0, sin 90 o = 1, sin 180 o = 0, sin 270 o = -1, sin 360 o = 0.

b. Cosines function

From the picture above , we obtain :

OQ = 1 when θ = 0® and OQ will decrease when θ increase, and the minimum OQ is

0 when θ = π/2. And then OQ increase but the value is negative and OQ is -1 when θ

= π.

c. Tangent Function

Tan θ value will be 0 if θ = 0 °

tanθ= sinθcosθ

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¿ 01

¿0

It will be heading towards + ∞ if θ= 90o because at that time also PQ +∞ and tan (-θ)

will lead to - ∞. So tan θ value between - ∞ to + ∞.

The value of tan θ = 1 when θ = 45o because at that time PQ = OQ

We can generalize it into :

D. Periodicity of Trigonometric Function

A function f is called periodic function if there are number nonzero p the element of

domain f such that f (x+p) = f(x) for every x element of domain f. Smallest Positive number

p which caused f(x+p)= f(x) for every x element of domain f is called periodic function f.

Periodic of Sinus and Cosines function is 360°.

Sin (θ°+k.360°) =sin θ°, k є {integers}

Cos (θ°+k.360°) =cos θ°, k є {integers}

Periodic tangent function is 180°.

Tan (θ°+k.180°) =tan θ°, k є {integers}

in generalization : periodic function f : x sin ax is 360o

|a| for a ≠ 0

periodic function f : x cos ax is 360o

|a| for a ≠ 0

periodic function f : x tan ax is 180o

|a| for a ≠ 0

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The proof of sinus periodic :

f : x sin ax0

f(x) = sin ax0

we know that sin ax0 = sin (ax + k. 360)0

= sin a( x+ k .360a

¿ for k € {...., -3, -2, -1, 0, 1, 2, 3, ....}

For a > 0, periodic of f : x sin ax0 is 360o

|a|and for a < 0, periodic of f : x sin ax0 is

360o

−a. Therefore, periodic of f : x sin ax0 for a ≠ 0 is

360o

|a| .

Example :

Determine periodic in the function below :

a. F:x sin x

b. F:x sin 2x

c. F:x cos 3x

Answer :

a. Sin ( x +k.360 )= sin x for k є { integers }, then periodic of F:x sin x is 360

( the smallest positive value of k.360 )

b. F:x sin 2x

F (x) = sin 2x

= sin (2x + k.360)

= sin 2(x+k.180 )

Then, the periode of the function is 180.

c. F: x cos 3x

F(x) = cos 3x

= cos (3x+k.360)

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= cos 3(x+k.120)

Then, the periode of the function is 120

E. Maximum and minimum value

Consider figure below

Sin a0 = yr

Cos a0 = xr

Y maximum value is r and y minimum is –r ( consider point A and C ). Therefore,

maximum value of sin a0 is rr

= 1 and minimum value of sin a0 is −rr

= - 1. The

Maximum and minimum value is reached for a = 90 + k. 360 and a = 270 + k.360 with

k is integers.

X maximum value is r and x minimum value is –r ( consider point B and D ).

Therefore, maximum value of cos a0 is rr

= 1 and minimum value of cos a0 is −rr

= - 1.

The Maximum and minimum value is reached for a = k. 360 and a = 180 + k.360 with

k is integers.

By considering explanation above, then we can conclude that -1 ≤ cos a ≤ 1 and 1 ≤

sin a ≤ 1 for a element R.

F. Graphic of Trigonometric Function

i. Graph function f(θ ¿

1) Sinus Function

y = sin x, for 0o ≤ x ≤ 360o

1. Sketching sinus function graph by using table

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To sketch the graph, we can use the table of function value.

First, we make the table function value f:x sin x for values of x such that value

of x can be determined.

xo Sin x0

0 0

30 ½

45 ½ √2

60 ½ √3

90 1

120 ½ √3

135 ½ √2

150 ½

180 0

210 -½

225 -½ √2

240 -½ √3

270 -1

300 -½ √3

315 -½ √2

330 -½

360 0

Then, graph the points (0,0), (30, ½ ), (45, ½ √2),...........,(360,0) at the Cartesius

graph.

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2. Sketching graph of sinus function by using unit circle

The steps are :

1) Make the specials angle, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240,

270, 300, 315, 330 and 360 in the circle. It can put the angles position

easier in the x-axis.

2) From the figure above, it can be made points projection of radius corner

which connect with special angles, make horizontal line parallel with x-

axis.

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3) put the values of special angles at the interval 0 ≤ x ≤ 360 in the x-axis.

Then make vertical lines parallel y-axis.

4) by determining intersection points between the lines in the previous steps,

then it can be made smooth curve, it can be obtained sinus function

graphic at the interval [0, 2π]

2) Cosines Function

To construct a cosine function graph using the unit circle, do the following steps :

1. Draw a circle with radius 1.

2. Draw the diameter of the circle that forms an angle of 30 °, 45 °, 60 °, 120 °,

135 °, and 150 ° to the x-axis.

3. Draw a horizontal line through the point of intersection between the diameter

of the circle you have created. The horizontal lines, from top to bottom, has

position 1, 1/2 √ 3, 1/2 √ 2, 1/2, and 0.. Thus, these lines can then be used as a

guide in painting cosine graph.

4. By using a horizontal guide lines that have been painted in step (2), plot the

points that fulfill the function y = f (x) = cos x.

5. Connect the dots have been plotted with a smooth curve.

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Second way : By Cartesian diagram

Make a graph of y = cos x with 0 o < x < 360 o

1. Make a table of the value of y and x

2. Plot the points in the Cartesian diagram

3. Connecting the plots become a curve.

X o 0 30 60 90 120 150 180 210 240 270 300 330 360

Y 1 ½√3 ½ 0 - ½ - ½√3 -1 - ½√3 -½ 0 ½ ½√3 1

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If considered, graphs cosine function has a shape similar to the graph of the function sine.

Why is that? Note that, cos x = sin (90 ° - x). Whereas sin (90 ° - x) = sin - (x - 90 °) = -

sin (x - 90 °). Thus obtained, cos x = - sin (x - 90 °). Therefore, the graph of the function

is the result of a shift cosine function graph-sin x to the right as far as 90 ° or π / 2.

3) Tangent Function

y = tan x, for 0o ≤ x ≤ 360o

1. Sketching sinus function graph by using table

To sketch the graph, we can use the table of function value.

First, we make the table function value f:x tan x for values of x such that value

of x can be determined.

Then, graph the points (0,0), (45,1 ), (135, -1),...........,(360,0) at the Cartesius

graph.

2. Sketching graph of sinus function by using unit circle

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The steps are :

Make the specials angle, 30o, 45 o, 60 o, 90 o, 120 o, 135 o, 150 o, 180 o, 210 o, 225 o,

240 o, 270 o, 300 o, 315 o, 330 o and 360 o in the circle. It can put the angles position

easier in the x-axis. Then lengthen until the radius intersects y-axis.

5) From the figure above, it can be made points projection of radius corner

which connect with special angles, make horizontal line parallel with x-

axis.

6) By determining intersection points between the lines in the previous steps,

then it can be made smooth curve, it can be obtained tangent function

graphic at the interval [0, 2π]

4) Cosecant Function

y = cosec x, for 0o ≤ x ≤ 360o

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x Cosec x

0 ∞

30 2

45 1,41

60 1,15

90 1

120 1,15

135 1,41

150 2

180 ∞

210 -2

225 -1,41

240 -1,15

270 -1

300 -1,15

315 -1,41

330 -2

360 ∞

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5) Secant Function

Make a graph of y = sec x with 0o < x < 360 o

1. Make a table of the value of y and x

2. Plot the points in the Cartesian diagram

3. Connecting the plots become a curve.

X o 0 30 60 90 120 150 180 210 240 270 300 330 360

Y 1 2/3√3 2 - -2 -2/3√3 -1 -2/3√3 -2 - 2 2/3√3 1

6) Cotangent Function

y = cotan x, for 0o ≤ x ≤ 360o

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ii. Graph function kF (θ) with R k, k ≠ 0, or k ≠ 1

If F (θ) is multiplied by any real number k (k ≠ 1, or k ≠ 0) then the graph of the

function F k (θ) can be obtained from the graph F (θ) by increasing y coordinates of

each point on the graph F (θ) with k times.

So if the only changes seen in the graph of the function k F (θ) of the graph of the

function F (θ) is:

1) If k> 1, the graph F (θ) increases dilated (enlarged) on the x-axis k times.

2) If 0 <k <1, the graph F (θ) shrink (reduced) k times the x-axis.

3) If - 1 <k <0, the graph F (θ) rotates 180 ° to the x axis and reduced k times.

4) If k <- 1, the graph F (θ) rotates 180 ° to the x axis and enlarged k times.

Example 1.

Draw a graph of the function{( x , y ) |y=2sin x } dengan domain {x |−2 π≤x≤2 π }. The graph of the function is done by drawing first a few points that lie on the graph of

the function.

X -2π − 23

π −π 12

π 0 12

π π 23

π 2π

Y 0 1 12√3 1 1

2√3

12

0-

12

12√3

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-2,5

-2,0

-1,5

-1,0

0,0

0,5

1,0

1,5

2,0

2,5

21 2

1

y

x

→ y = sin x

→y = 2sin x

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

21 2

1

yx

21

→ y = cos x

Here k = 2, so k> 1 when compared with the graph of y = sin x then the graph of y = 2 sin

x, the x-axis increases, being a point y can be obtained by multiplying by 2.

Example 2

Draw chart {( x , y ) |y= 12

cos x } dengan domain {x |−2π≤x≤2π }. 

In this example k = 12

, so 0 <k <1. Compare the graph of y = cos x and y = ½ cos xn

seen here that the graph of y = ½ cos x shrink or minimized ½ times the x-axis.

Example 3

Make a graph of function

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-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

21 2

1

y

x21

→ y = tan x

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

21 2

1

y

→ y = -2 cos x

→ y = cos x

{( x , y ) |y=− 12

tan x } dengan domain {x |− 32

π<x< 32

π , x≠±12

π }In the picture below, k

= -12

, - 1 <k <0 can be seen that the graph of y = - 12

tan x is in the opposite direction on

the x-axis with the graph of y = tan x or rotates 180 ° to the x axis. As for the reduced

form ½ times the x-axis.

Example 4

Make a graph of the function{( x , y ) |y=−2 cos x } dengan domain { x |−2 π≤x≤2 π } . In this example k = - 2 so k <-1. It can be seen that the graph of y = - 2 cos x is in the

opposite direction to the x-axis with the graph of y = cos x or rotates 180 ° to the x axis.

As for the shape of the graph over 2 times the x-axis.

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x

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

y

x

21

→ y = sin x

iii. Graph function F (mα), m real numbers m ≠ 0 or m ≠ 1

F If the angle is multiplied by any real number m, trigonometric functions here

denoted by F (mα).

Graph of the function F (m) can be classified into two parts:

1) If m <0 then the period of the function F (mα) is 1

|m|time from the period of F (α ).

2) If m> 0 then the period of the function F (mα) is 1m

time from the period of F (α).

Example 1

Make a graph function{( x , y ) |y=sin 12

x } dengan domain { x |−2 π≤x≤2 π }.

From the figure above shows that the period of the function y = sin 12

x is 2 x 2π or 2

times the period of sin x. so for the function F (12

x) times the period of the period 2 F (x).

Example 2

Make a graph of the function {( x , y ) |y=sin (−2x ) } dengan domain {x |−π≤x≤π }.Period of the function sin (-2x) is or ½ x 2 ½ times the period of sin x. This means that

the period of sin (- 2x) is

1|−2| time function sin x. because sin (- 2x) = - sin 2x then the

graph of y = sin (- 2x) is also the graph of y = - sin 2x. Thus the graph of y = sin (- 2x) is

located opposite to the direction of the x-axis with the graph of y = sin 2x.

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-1,0

0,0

1,0

49 2

3 21 2

1 23

y

x

→ y = sin x

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iv. The function F (α +θ ) is a graph with θ as Constant Angle

Graph of the function F (α +θ ) can be obtained by translating (shift) at intervals

along the axis as |θ|.

1) If> 0 interval on the axis is translated to the left along |θ|

2) If <0 interval on the axis is translated to the right along |θ|Example 1

Make a graph of the function {( x , y ) |y=sin (x+ 1

4π )} dengan domain {x |− 9

4π≤x≤ 7

4π }.

 

Seen that the graph of y = sin (x +14

π ) is the same shape with the graph of y = sin x, but

moved located to the left 14

π .

Example 2

Draw chart of function {( x , y ) |y= 12

cos2( x− 12

π )} dengan domain { x |−2π≤x≤2π }.

To draw a graph of the function can be done in two ways:

a.draw a point on the graph according to the domain, then the points are connected by a

smooth curve.

b. By comparing the graph cos x.

If done in a second so we can do as a graph is a trigonometric function with form

kF [m (α +β )].

1) θ=−12

π means> 0 graphs can be obtained from the graph cos x by translating the

interval to the right along the x axis.

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-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

23 2

1 21 2

3

y

x → y = cos x

2) k = ½, 0 <k <1, then the graph can be obtained from the graph cos x to minimize 12

times the x-axis.

3) m = 2, then m> 0, then the period of the graph is 12

times of the period graph cos x =

π .

v. Graph function F (α) = g (a) + f (α), g (a) = Constant Function.

If f (α) is a trigonometric function of angleα , g (a) refers to the constant functions are

the domain of g and f equal, then to draw the function F (α ) = g (a) + f (α) is

calculated by summing the coordinates of each point of both graphics of the

corresponding domain.

Example 1

Draw a chart of function{( x , y ) |y=2+sin x } dengan domain { x |−2 π≤x≤2 π } .

21

-1,0

0,0

1,0

2,0

3,0

23 2

1 21 2

3 y

x

→ y = sin x

→ y = 2 + sin x

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REFERENCES

http://ceritabersama-tati.blogspot.com/2012/06/grafik-fungsi-trigonometri.html

http://www.scribd.com/doc/45279202/GRAFIK-FUNGSI-TRIGONOMETRI

http://yos3prens.wordpress.com/2013/02/26/melukis-grafik-fungsi-cosinus/

http://guru-indonesia.net/admin/file/f_11423_ARTIKELMENGGAMBAR

GRAFIKFUNGSITRIGONOMETRI.pdf

http://id.scribd.com/doc/45279202/GRAFIK-FUNGSI-TRIGONOMETRI

http://www.regentsprep.org/Regents/math/algtrig/ATT7/othergraphs.htm

Prayitno, Budhi. 1997. Matematika untuk SMU jilid 2A. Jakarta: Erlangga

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