student book trigonometry

8/3/2019 Student Book Trigonometry

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TRIGONOMETRIC RATIOS

AND FUNCTION

Intelligent Learning

Mathematics

MOTIVATION:

Trigonometric geometry is very iseful in daily life. It can

helpas in determining distance of two points, length/height

of tower, height of tree, and position of sailing ship with

observation distance towards certain elevation angle. Using

this science, complex problems can be made simple.

NOOR AZIZAH



The reffered unit circle is circle with the

length of radius q unit and control is point O.

From the figure, the formula for

trigonometry comparison for angle is as

follows.

To study these trigonometric functions are reminded prerequisite knowledge

understands the function. From the understanding developed understanding of the

function f is a function of the trigonometric functions on real numbers , for is a trigonometric ratio which has been described before.

Examples: a. Function Sine

b. Function Cosine

c. Function Tangent

The graph of trigonometric function, like function sine, cosine, or tangent can

be drawn by using unit circle and trigonometric table for extraordinary angles .

1. Drawing Graphs Using Unit CircleBefore you can fully understand the behavior of any

sort of function, you have to graph it. In the case of the

trig functions, the graph is pretty easy to draw if you look

at the unit circle for a while. The strategy is

simple: start out at , and follow the unit

circle around, watching what the y -coordinate

or x -coordinate.

Draw Simple Trigonometric Function Graphs



a. The graph of , for

b. The graph of , for

c. The graph of , for

Kochansky’s Way: How to draw approaches the value of π.

Draw EF = 3r

So that,

Aapproaches the value of π ≈

3,141592635897932384626433832795.

The graph of the trigonometricfunction , for can described as

follows.



Doing exercises on Exercise 1 on the worksheet individually!

Draw the graph of function , for !

Solution:

Sin 0o

is zero, as you hopefully figured out for yourself by looking

at the unit circle last page. Sin 90o

is 1. In between, as moves

from zero to ninety, the y-coordinate on the unit circle is

constantly increasing. So the graph starts out like:

Example

Continuing to follow the unit circle around, as increases to 180o

the y-coordinate drops

to zero: we're now at the point (-1,0) on the unit circle. As keeps going from there, the

y-coordinate doesn't turn around and go back up to one; it keeps going down, reaching -1

when gets to 270o. Then the y-coordinate turns around, and reaches zero again when

is 360o; we are now, of course, at exactly the same place on the unit circle that we were at

when was 0o. So the full sine graph looks like:

You can keep on going from there around the unit circle, retracing the path around and

around; so the graph repeats itself indefinitely.

Check Understanding:



2. Drawing Graphs Using Extraordinary Angles

Other than utilizing unit circles, function graphs for sines, cosine, and tangent

can also be drawn by initially making a table for function values for

extraordianry angles.

In additon, to draw a graph of trigonometric function, it is necessary to take

notice of the trigonometry function value of the extraordinary angles. By

subsituting the extraordinary angles, the graph of the trigonometric function , for can described

as follows.


Draw the graph of function , for !

Solution:

The values of for extraordinary angles are shown in the followingtable.

F(x) 0o 30o 60o 90o 120o 150o 180o 210o 240o 270o 300o 330o 360o

Cos

x1 1/2

1/2 0 -1/2

-

1/2 -1

-

1/2 -1/2 0 1/2 1/2

1

From the table, the point pairs of (x, f(x)) or (x,y) , which are (0o,0), (30o, 1/2 ), (60o,

½), (90o,0), (120o, -1/2), (150o, -1/2 ), (180o,-1), (210o, -1/2 ), (240o-,0), (300o,1/2), (330o, 1/2 ), and

(360o,1). Further, the pairs of the point are drawn in Cartesian coordianate by

smooth curve so that the graph of function , for isobtained.

Example




3. Amplitude, Period, and Phase Shift

Amplitude

How do the function and compare with the

function ? We will make a table of values and sketch the curves.

In the following tables, approximate values of irrationa values of the sine

function are used.

When we use these values to sketch the curves , , and , we see that is the function stretched in

the vertical direction and

is the function

compressed

in the vertical direction as expected.For , the maximum function value is 1 and the minimum

function values is -1.

For , the maximum function values is 2 and the

minimum funtion values is -2.

For , the maximum function values is ½ and the

minimum funtion values is -1/2.

For the function , the maximum function values is |a| and

the minimum funtion values is -|a|.



For the function , the maximum function values is |a|

and the minimum funtion values is -|a|.

The amplitude of periodic function is the absolute value of one-half the

difference between the maximumand minimum een the maximumand

minimum y -values.

For , the amplitude is .



In general:

For the function and , the amplitude is Period

The function is the function stretched or compressed

by a factor a in the horizontal direction. Compare the graphs of , , and . Consider the maximum, zero, and minimum

values of y for one cycle of the graph of

.

The graph shows the functions

(---),

(---), and

(---) in the interval 0 ≤ x ≤ 4π. The graph of is the



graph of compressed by the factor ½ in the horizontal direction.

The graph of

is the graph of

stretched by the factor 2

in the horizontal direction.

For , there is one complete cycle in the interval 0 ≤ x ≤

2π.


π.


4π.

The difference between the x -coordinates of the endpoints of the interval

for one cycle of the graph is the period of the graph.

The period of is 2π.

The period of is π.

The period of is 4π.

In general:

The period of and is .Phase Shift

The graph of is the graph of moved |c| units to the right when c is

negative or |c| units to the left when c is possitive. The horizontal of a

trigonometric function is called a phase sift. Compare the graph of and

the graph of .



The graph of is the graph of moved π/2 units to the left.


4. Sketch Graph Using Software

In addition to the above two ways, there are also other ways to draw a graph

trigonometry, using software, such as MAPLE. This method can provide its own

motivation for students.

Spiral Archimedes: for as a constanta.

Spiral Logaritmic , a constanta




Cardioda: , a contant and 0 ≤ ≤ 2π

Limason: , a,b constant and 0 ≤ ≤ 2π

Try practicing these functions on your computer at home, are you able to get

similar results to the picture above!




Competen

1. Choose the graph of , for - 2π ≤ x ≤ 2π.

a. c.

b. d.

2. Which of the following is True for the function y = cos x?

a. Even function c. Odd and even function

b. Odd funtion d. Cannot be determined

3. Choose the graph of

, for 2π ≤ x ≤ 2π.

a. c.

b. d.

4. For what value of the following the function y = - 8 sec x is undefined?

a. 41/4 π c. 31/3 π

b. 21/2 π d. 11 π

5. What is the period of the function y = - 5/4 sec 20x ?

a. 1/20 π c. 10π

b. 20/π d. 1/10 π

Competence Test:



6. Given the following graph, aanser question a-g.!

a. Indicate a cycle on this graph by using trick marks (or any other method)

to indicate the beginning and end of the cycle.

b. What is the period of this function?

c. What is the frequency of this function?d. What is the amplutude of this function?

e. Which of the following equations must be associated with the above

graph?

f. Is this an even or an odd function?

g. What is the range of this function?

7. Sketch the graph of y = - sec 2x for .

8. Sketch the graph of y = 1 2csc x, for - 2π ≤ x ≤ 2π, and find its period and

amplitude!

9. Sketch the graph of , a is constant and 0 ≤ x ≤ 2π by using Software!

10. Suppose Tarzan is swinging back and forth on his grapevine. As he swings, hegoes back and forth over the river bank, hovering alternately over land andwater. Jane decides to mathematically model his motion and starts her stopwatch. Let t be the number of second the stopwatch reads and let y be the

number of meters Tarzan is from the riverbank (not his height). Assume that y varies sinusoidally with t and that y is positive when Tarzan is over water

and negative when Tarzan is over land. Jane finds that at 2 seconds, Tarzanis at one end of his swing 23 feet from the riverbank over land. At 5 seconds,Tarzan is at the other end of his swing 17 feet from the riverbank over water.Generate the equation that represents Tarzan’s swing model



1. Basic Identities

You will recall that an identity is a statement which is always true. In contrast, an

equation is a statement which is only true for certain values of the variables involved.

For example,

We already know some identities. Some are definitions. Other have been proven. We

begin by listing all the identities we should know.

Prove Simple Trigonometric Identity

Are equations; they are only

true for certain values of x .

is an identity; it is true no

matter what x and y are.



Pythagorean Identities

1. 2.

3.

Reciprocal Identities

1.

2.

3.

4.

5.

Reciprocal Identities

1.

2. –

3.

Confuction Identities

1.

2.

We already proven all these identities, except the conjuction identities. We have

already mentioned them when we studied tranformations of the graphs of sine and

cosine.

These identities will be used as our starting point for proving more identities. Before

we do this, you may have already asked yourself: what are identities used for? One

answer is the that learning how to prove identities is a good exercise for the brain.

But identities are useful for other reasons. Very often, identities allow you to simplifyexpressions. The simpler an expression is, the easier it is to work with. Identities are

also used in solving trigonometric equations.



Consider again the trigonometric ratio of the material below, try to discuss the

using of the ratios in proving the basic identities!

Doing the Exercise 1 on worksheet!

2. Proving an Identity

The eight basic identities are used to prove other identities. To prove an identity

means to show that the two sides of the equation are always equivalent. It is

generally more efficient to work with the morw complicates side of the identity and

show, bynusing the basic identities and algbraic principles, that the two sides are the

same.

Use the basic identities to show that for all values of x for which each

side of the equation is defined.

Solution:

Each of the functions in the given equation can be written in terms of , , or both.

1. Use the basic identities to write each side of the identity in terms of and.

2. Divide a numerator and a denominator of the left side of the equation by :

Note: if csc x is defined,

Example

Discussion:



a. Sum and Differenc e Identities

Prove that is an identity.

Solution:

Write the left side of the equation in terms of and .

Proof begins with what is known and proceeds to what is to be proved. Although

we have written the proof in Example by starting with what is to be proved and

ending with what is obviously true, the proof of thihs identity really begins with the

obviously true statement.

Example 1

1

Prove the identity .

Solution:

For this identity, it appears that we need to multiply both sides of the equation by to clear the denominator. However, in proving an identity we perform

only operations that change the form but not the value of the side of the equation.

Continue the completion of these examples!

Example 2

1



Is an identity? Explain why or why not?

For what values of x is the identity undefined?

3. Other Identities

Sum and Difference Identities

1.

2.

3.

4.

5.

6.

Double and Half-Angle identities

1.

2.

3.

4.

5.

6.

Discuss with your friends to prove the above identities!

Discussion:

Discussion:



Competen

1. Knowing that , calculate the remaining trigonometrios of angle x .

2. By determining trigonometric ratios values of extraordinary angles, prove

the following statements.

a.

b.

c.

3. We have and , where x and y are acute angles.

Determine the following values.

a.

b.

c.

4. Use the pythagorean identities to write:

a. in terms of

b. in terms of

5. Prove the identities:

a.

b.

c.

d.

Competence Test:



Siswanto. 2009. Theory and Application of Mathematics. Solo: Tiga Serangkai.

Laval, Philippe B. Trigonometric Identities. Kennesaw State University.

http://www.intmath.com

http://www.purplemath.com

http://www.analyzemath.com/unitcircle/unitcircle.html

http://www.onlinemathlearning.com/trig-graphs.html

http://roslynschools.org/schools/amsco/algebra2/Chapter12.pdf


http://www.belajar-matematika.com

http://www.intmath.com/


http://www.purplemath.com/










http://www.belajar-matematika.com/











Draw Simple Trigonometric

Function Graphs

1). b (Graph 3)

3). c (Graph 1)

5). d (-1/10 π)

7). Make a table of values for ordered

pairs of the form (x, - sec 2x).

Plot the points and connect them

with a smooth curve. Draw dashed

vertical lines at the points where the

function y = - sec 2x is not defined.

Notice that the function repeats at

intervals of π units . Therefore, theperiod of y = - sec 2x is π. The

function increases without bound

over each interval, so the amplitude

is not defined.

9).

1). First quadrant:

Second quadrant?

3). a (117/44)

b (452/576)

c (75/125)

5). a (proven)

b (proven)

c (proven)

d (proven)

Prove Simple Trigonometric

Identity

student book trigonometry

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