trigonometric functions: the unit circle 1.2. objectives students will be able to identify a unit...

Trigonometric Functions: The Unit

Circle1.2

Objectives Students will be able to identify a unit circle

and describe its relationship to real numbers.

Students will be able to use a unit circle to evaluate trigonometric functions.

Students will be able to use the domain and period to evaluate sine and cosine functions.

Students will be able to use a calculator to evaluate trigonometric functions.

The Unit Circle

(1, 0)

(0, 1)

(0, -1)

(-1, 0)

Definitions of the Trigonometric Definitions of the Trigonometric Functions in Terms of a Unit CircleFunctions in Terms of a Unit Circle

If If tt is a real number and (x, y) is a is a real number and (x, y) is a point on the unit circle that point on the unit circle that corresponds to corresponds to tt, then, then

sin t y cos t x tan t y

x, x 0

csc t 1

y, y 0 sec t

1

x,x 0 cot t

x

y, y 0

Use the Figure to find the values of the trigonometric functions at t=/2.

/2/2

(1,0)

(0,1)

x2 y2 1

Solution:

The point P on the unit circle thatCorresponds to t= /2 has coordinates(0,1). We use x=0 and y=1 to find theValues of the trigonometric functions

sin2

y 1 cos2

x 0

csc2

1

y

1

11 cot

2

x

y

0

10

tan t yx

and sec t 1x

10

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THE DOMAIN AND RANGE OF THE SINE AND COSINE FUNCTIONS AND THEIR PERIOD

The domain of the sine function and the cosine function is the set of all real numbers

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

The period is 2π. This means it repeats every Periodic: f(t+c)=f(t) where c= 2π. Page 152 # 36, 42

Definition of a Periodic Function

A function f is periodic if there exists a positive number p such that

f(t + p) = f(t)

For all t in the domain of f. The smallest number p for which f is periodic is called the period of f.

Periodic Properties of the Sine and Cosine Functions

sin(t + 2) = sin t and cos(t + 2) = cos t

The sine and cosine functions are periodic functions and have period 2.

Periodic Properties of the Tangent and Cotangent Functionstan(t + ) = tan t and cot(t + ) = cot t

The tangent and cotangent functions are periodic functions and have period .

Even and Odd Trigonometric Functions

The cosine and secant functions are even.cos(-t) = cos t sec(-t) = sec t

The sine, cosecant, tangent, and cotangent functions are odd.

sin(-t) = -sin t csc(-t) = -csc t

tan(-t) = -tan t cot(-t) = -cot t

EXAMPLE Use the value of the trigonometric function at t

= /4 to find sin (- /4 ) and cos(- /4 ).

sin4

2

2cos

4

2

2tan

4

1

csc4

2 sec4

2 cot4

1

Solution:

sin(-t) = -sin t, so sin(- /4 ) = -sin(/4 ) = -2/2cos(-t) = cos t, so cos(- /4 ) = cos(/4 ) = 2/2

Try it: Pg. 151 # 38, 40, 46, 48, 50, 56

Homework: # 5 – 57 odd

Right Angle Trigonometry

Objectives

Students will be able to evaluate trigonometric functions of acute angles.

Students will be able to use fundamental trigonometric identities.

Students will be able to use a calculator to evaluate trigonometric functions.

Students will be able to use trigonometric functions to model and solve real life problems.

The Six Trigonometric Functions

HypotenuseSide opposite .

Side adjacent to .

The figure below shows a right triangle with one of its acute angles labeled . The side opposite the right angle is known as the hypotenuse. The other sides of triangle are described by the position relative to the acute angle . One side is opposite and one is adjacent to .

HypOpp

Adj

HypOppsin

HypAdj

cos

AdjOpp

tan

OppHyp

csc

AdjHyp

sec

OppAdj

cot

RIGHT TRIANGLE DEFINITIONS OF TRIGONOMETRIC FUNCTIONS

How does compare to the unit circle? Page 160 #8, 12

45°

45°

1

1√2/2

30°

60°

1 1

2 2

√3

Reciprocal IdentitiesReciprocal Identities

sin t 1

csctcos t

1

secttan t

1

cot t

csc t 1

sin tsec t

1

cos tcot t

1

tan t

Quotient Identities

tan t sint

cos tcot t

cos t

sin t

sin2 t cos2 t 1

1 tan2 t sec2 t

1 cot2 t csc2 t

sec)90(sc csc)90sec(

tan)90(ot cot)90tan(

sin)90cos( cos)90sin(

oo

oo

oo

c

c

Examples

• Page 161 #32, 38, 44, 46, 58, 62, 66, 70

• Homework: 5 – 47 odd, 57 – 67 odd