TOPIC 1.1 - TRIGONOMETRIC TOPIC 1.1 - TRIGONOMETRIC FUNCTIONSFUNCTIONS

1.1.1: Angles and Their Measure1.1.2: Right Triangle Trigonometry1.1.3: Computing the Values of Trigonometric Functions of Acute Angles1.1.4: Trigonometric Functions of General Angles1.1.5: Unit Circle Approach; Properties of the Trigonometric Functions 1.1.6: Graphs of the Sine and Cosine Functions1.1.7: Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions

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1.1.1: Angles and their 1.1.1: Angles and their measuremeasure

Angles:An angle is formed by initial side and terminal sideThe common point for this 2 sides is the vertex of the angleAn angle is in standard position if:

i) Its vertex is at the origin of a rectangle coordinate systemii) Its initial side lies along the positive x-axisPositive angles generated by counterclockwise rotationNegative angles generated by clockwise rotationAn angle is called a quadrantal angle if its terminal side lies on the x-axis or the y-axis

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Measuring angles using degreesMeasuring angles using degrees360

90

1 60

1 60

• One complete revolution =

• One quarter of a complete revolution =

• One degree equals 60 minutes, i.e.

• One minute equals 60 seconds, i.e.

.

= one right angle

Angles classified by their degree measurement:

a.Acute b. right c.Obtuse d.Straight angle angle angle angle

3

2

Measuring angles using radians

•One complete revolution = radians

• One radian is the angle subtended at the center of a circle by an arc of the circle equal in length to the radius of the circle.

c

180

1 1

180c

radians

4

radians

3

4

Example:Convert each angle in degrees to radians a. 60° b. 270° c. -300°Convert each angle in radians to degrees a. b. c. 6 radians

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1.1.2: Right Triangle 1.1.2: Right Triangle TrigonometryTrigonometry

The six trigonometric functionFor any acute angle of a right angled triangle OAB (figure shown)

sin Opposite

Hypotenuse

b

ccos Adjacent

Hypotenuse

a

ctan Opposite

Adjacent

b

a

cossin

ec

1 seccos

ant

1cot

tanangent

1

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Fundamental identitiesFundamental identities

csc

1sin

sec

1cos

cot

1tan

sin

1csc sec

cos

1

cottan

1

Reciprocal identities

tansin

cos

cot

cos

sin

Quotient identities

sin cos2 2 1

22 sectan1 1 2 2 cot csc Pythagorean identities

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Trigonometric functions Trigonometric functions and complementsand complements

)90cos(sin )90sin(cos )90cot(tan

)90tan(cot )90csc(sec )90sec(csc

Cofunction identitiesThe value of a trigonometric function of of the complement of

is equal to the cofunction

46sin12

cot

Example: Find a cofunction with the same value as the given expression a) b)

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Solving Right TrianglesSolving Right Triangles

180

90

To solve a right triangle means to find the missing lengths of its sides and the measurements of its angles.Some general guidelines for solving right triangles:1. Need to know an angle and a side, or else two sides.2. Then, make use of the Pythagorean Theorem and the fact that the sum of the angles of a triangle is in a right triangle is

.

, and the sum of the unknown angles

c

a

b

A

222 bac 222 bac

c2=a2+b2 A+B=90°

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Example:Example:If b = 2 and , find a, c, and .

Solution :

40

2

c

a

40

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1.1.3: COMPUTING THE VALUES OF 1.1.3: COMPUTING THE VALUES OF TRIGONOMETRIC FUNCTIONS OF ACUTE TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLEANGLE

30 60 45

We use isosceles triangle and equilateral triangle to find these special angles of

, and

sin603

2 cos60

1

2 tan60 3

sin301

2 cos30

3

2 tan30

1

3

sin451

2

2

2 cos45

1

2 tan45 1

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1.1.4: TRIGONOMETRIC FUNCTIONS OF 1.1.4: TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLESGENERAL ANGLES

1st Quadrant2nd Quadrant

3rd Quadrant 4th Quadrant

Definitions of trigonometric functions of any angle

22 yxr

Let be any angle in standard position, and let P = (x,y) be a point on

. If

is the distance from (0,0) to (x,y), then the 6 trigonometric functions of

are defined by the following ratios:

the terminal side of

r

ysin r

xcos 0,tan x

x

y

0,csc yy

r 0,sec xx

r 0,cot yy

x

x

ry

x

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Example: 1.Let P = (4, -3) be a point on the terminal side of

six trigonometric functions of

2. Evaluate, if possible the cosine function and the cosecant function at

. Find each of the

the following 4 quadrantal angles

2

2

3

a)

= 180º

=

d) =

b)= 0°

c)

**Quadrantal Angles: 0 °,90°,180°,270°,360°

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The signs of the trigonometric functionThe signs of the trigonometric function

x

y

All (sin , cos, tan)sine

cosinetangent

If

depends on the quadrant in which lies

is not a quadrantal angle, the sign of a trigonometric function

Example:Given tan = -1/3 and cos < 0, find sin and sec

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2. The trigonometric functions of coterminal angles are equal.Example:

Coterminal AnglesCoterminal AnglesTwo angles in standard position are said to be coterminal if they have the same terminal side.Example:

For example, the angles 60° and 420° are coterminal, as are the angles -40 and 320°.Note:

k 2

)2sin(sin k

1. is coterminal with , k is any integer.

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Definition of a reference angle

Let

Its reference angle is the positive acute angle ´ formed by the terminal side of

and the x-axis

be a nonacute angle in standard position that lies in a quadrant.

Example:1.Find the reference angle, ´ for each of the following angles:

4

7

a) b) =

c) d) = 3.6

= 210º

= -240º

2. Use the reference angles to find the exact value of the following

trigonometric functions:

4

5tan

6sec

a) sin 300º b) c)

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1.1.5: UNIT CIRCLE APPROACH; PROPERTIES 1.1.5: UNIT CIRCLE APPROACH; PROPERTIES OF THE TRIGONOMETRIC FUNCTIONS OF THE TRIGONOMETRIC FUNCTIONS

Definitions of the trigonometric functions in terms of a unit circleIf t is a real number and P = (x,y) is a point on the unit circlethat corresponds to t, then

yt sin xt cos 0,tan xx

yt

0,1

csc yy

t 0,1

sec xx

t 0,cot yy

xt

tExample: Find the values of the trigonometric function at

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all real numbers. The range of these functions is the set of all real numbers from -1 to 1, inclusive.

The domain and range of the sine and cosine functionsThe domain of the sine function and the cosine function is the set of

tt cos)cos( tt sec)sec(

Even and odd trigonometric functionsThe cosine and secant functions are even

tt sin)sin(

tt csc)csc(

tt tan)tan(

tt cot)cot(

The sine, cosecant, tangent and cotangent functions are odd

)60cos(

6tan

Example: Find the exact value of: a) b)

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Definition of a periodic functionA function f is periodic if there exists a positive number p such that

)()( tfptf for all t in the domain of f. The smallest number p for which f is periodicis called the period of f

tt sin)2sin( tt cos)2cos( 2

Periodic properties of the sine and cosine functions

and

The sine and cosine functions are periodic functions and have period

tt tan)tan( tt cot)cot(

Periodic properties of tangent and cotangent functions

andThe tangent and cotangent functions are periodic functions andhave period

tnt sin)2sin( tnt cos)2cos( tnt tan)tan(

Repetitive behavior of the sine, cosine and tangent functionsFor any integer n and real number t,

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1.1.6: Graphs of the Sine 1.1.6: Graphs of the Sine and Cosine Functionsand Cosine Functions

1 1y

Characteristics of the Sine Function:

Domain : all real numbers Range :

2 sin)sin(

Period :Symmetry through origin :

Odd function x - intercepts : ...., , , , , , ,...... 2 0 2 3 y - intercept : 0

x ....., , , ,...3

2 2

5

2

x ....., , , ,... 2

3

2

7

2

max value : 1 , occurs at

min value : -1 , occurs at

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Graphing variations of y=sin xGraph of y=A sin Bx1.Identify the amplitude and the period

Amplitude = |A| ; Period = B

2

2. Find the values of x3. Find the values of y for the one that we find in step 24. Connect all the points and extend to the left or right as desired

Graph of y = A sin (Bx – C)This graph is obtained by horizontally shifting the graph of y=A sin Bx so that the starting point of the cycle is shifted from x = 0 to

B

Cx

This is called the phase shift

If 0B

C the shift is to the right

If 0B

C the shift is to the left

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Example: 1- Determine the amplitude of y = 3sin x. Then graph y = sin x and y = 3sin x for

20 x

xy sin2

1 xy sin xy sin

2

1

3 x

2- Determine the amplitude of . Then graph and

for

xy2

1sin2

80 x

3- Determine the amplitude and period of . Then graph the function

for

4- Determine the amplitude, period, and phase shift of 32sin3 xy

Then graph one period of the function

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Characteristics of the Cosine Function:

Domain : all real numbers

1 1y2

cos( ) cos

Range :

Period : Symmetry about y-axis :

Even function

x - intercepts : ....., , , , , ,... 3

2 2 2

3

2

5

2

y - intercept : 1

x ..., , , , ,......2 0 2 4

x ...., , , , ,...... 3 5

max value : 1 , occurs at

min value : -1 , occurs at

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Graphing variations of y=cos xGraph of y=A cos Bx1.Identify the amplitude and the period

Amplitude = |A| ; Period = B

2

2. Find the values of x3. Find the values of y for the one that we find in step 2

4. Connect all the points and extend to the left or right as desired

B

Cx

Graph of y = A cos (Bx – C)This graph is obtained by horizontally shifting the graph of y=A cos Bx so that the starting point of the cycle is shifted from x = 0 to This is called the phase shift

If 0B

C the shift is to the right

If 0B

C the shift is to the left

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xy cos422 x

Example: Determine the amplitude and period of

Then graph the function for

Vertical shifts of sinusoidal graphsFor y = A sin (Bx – C) + D and y = A cos (Bx – C) + D, the constant +D will cause the graph to shift upward while –D will cause the graph to move downward.

So, the max y is D + |A| and the min y is D - |A|

Example: Graph one period of the function y = 2 cos x + 1

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1.1.7: GRAPHS OF THE TANGENT, 1.1.7: GRAPHS OF THE TANGENT, COTANGENT, COSECANT, AND SECANT COTANGENT, COSECANT, AND SECANT FUNCTIONSFUNCTIONS

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Characteristics of the Tangent Function:

Domain : all real numbers except odd multiples

tan( ) tan

Range : all real numbersPeriod :

Symmetry with respect to the origin :

Odd function

x ..., , , , , , ,......2 0 2 3

x ....., , , , ,...3

2 2 2

3

2

x - intercepts :

y - intercept : 0 Vertical asymptotes :

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Characteristics of the Cotangent Function

Domain : all real numbers except integral multiples of Range : all real numbersPeriod :

cot)cot( ......2,,0,2,..., x

...2,,0,....., x

Symmetry with respect to the origin : x - intercepts : y - intercept : noneVertical asymptotes :

Odd function

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Characteristics of the Cosecant Function:

1y 1y

2

Domain : all real numbers except integral multiples of Range : all real numbers of y such that or Period :

csc)csc(

...2,,0,....., x

Symmetry with respect to the origin : x - intercepts : noney - intercept : noneVertical asymptotes :

Odd function

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Characteristics of the Secant Function:

2

1y 1y2

Domain : all real numbers except odd multiples of Range : all real numbers of y such that or

Period :

sec)sec(

x ....., , , , ,...3

2 2 2

3

2

Symmetry with respect to y-axis: x - intercepts : noney - intercept : 1Vertical asymptotes :

Even function

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