Lecture 1.41 Value of Trig Functions

8/7/2019 Lecture 1.41 Value of Trig Functions

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0.2: Trignometric FunctionsObjectiv

e:•Understandrelationships between

angles and sides of righttriangles

•Review trig functions of ©2008 Roy L. Gover(www.mrgover.com)

Calculus PrerequisiteKnowledge



Definition

The special angles are:•30°,60°, and 45° in

degrees

• and in radians

, ,

6 3 4

π π π

In calculus, we use

radians.



Important Idea

These angles arespecial because theyhave exact value trig

functions.



Consider the first two

special angles…

Long

S h o r

t

S i d e

H y p o t e n u s

e 6π

3π



L

o n g S i d e

H y p o t e n

u s e 6

π

3

π

Short Side…

orientationdoes not

change therelationshi

psbetween

sides and



L o

n g S

i d e H y

p o t e n

u s e 6

π

3

π

Short Side

…

orientation does not


ps betweensides and

angles



Analysis

Consideran

equilateral

trianglewithsides 2

2

2

2



Analysis

...equilateral triangles

areequiangula

r and eachangle must

2

2

2

3

π

3π

3π

3π



Analysis

…theangle

bisectorintersects

the baseat themidpoint

22

1 1

3π

3π

6

π

6

π



Analysis

2 2

1 1

1

2?

?= 3

6

π

6

π

6

π

3π

3π

3

π



Try This

Find theheight of

anequilateral

trianglewith eachside 4

4 4

4

?



Important Idea

In a rad triangle:, ,6 3 2

π π π

•the short side is one-half the hypotenuse.

•the long side istimes the short side.3 M

e m o r i z

e



Try This

Find thelength

of themissing

sides:

6

π

10

5

5 3



Try This

Find thelength

of themissing

sides:

3

π

5

5 3

3

10 3

3



Try This

What iswrong

with thispicture? 6

π



Consider the last special

angle…

6

π

3

π

4

π

4π

4

π



H y p o t e

n u s e

…

orientationdoes not


psbetween

sides and

4

π

4

π



H y p o t e n

u s e

…

orientationdoes not


psbetween

sides and

4

π

4

π

d



…andsidesoppositeequal

anglesare

equal... 4

π

4

π x

x

and by the

2 x

pythagorean theorem,

the hypotenuse is



Important Idea

In a triangle:

• The legs of thetriangle are equal.

•the hypotenuse istimes the length of theleg.

M e m

o r i z

e2

, ,4 4 2

π π π



Try This

Findthe

lengthof themissing

4

π

4

π

2

22 2



Try This

Findthe

lengthof themissing

2

2

24

π

4

π



Review

•

-1 1

-1

1

x

yr

θ

oppositesin

hypotenuse

y

r θ = =

adjacentcos

hypotenuse

x

r θ = =

oppositetan

adjacent

y

x

θ = =

•

• Memoriz



Review

•

-1 1

-1

1

x

yr

θ

hypotenusecsc

opposite

r

yθ = =

hypotenusesec

adjacent

r

xθ = =

adjacentcot

opposite

x

y

θ = =

•

• Memoriz



Find theexact

values of the sixtrig

3

π - 1 1

- 1

1Example

functions of thegiven special

angle in standard



- 1 1

- 1

1

Analysis

1. Make

a righttriangle.

θ

rads3

π θ =





- 1 1

- 1

1

Analysis

32

1

3. Use thedefinitions

of the trigfunctions to

find eachtrig function

exact value

θ

rads3

π θ =



- 1 1

- 1

1

Analysis

32

1

opp 3sinhyp 2

θ = =

adj 1cos

hyp 2θ = =

opp 3tan 3

adj 1θ = = =

θ

rads3

π θ =



- 1 1

- 1

1

Analysis

32

1

hyp 2 3cscopp 3

θ = =

hypsec 2

adjθ = =

adj 3cot

opp 3θ = =

θ

rads3

π θ =



Find theexact

values of the sixtrig

- 1 1

- 1

1Try This

functions of thegiven special

angle in standard

4π



Solution

2sin4 2

π =

2cos

4 2

π =

tan 14

π =

csc 24

π =

sec 24

π =

cot 14

π =



Important Idea

If the given angle isother than a quadrant Iangle, the exact valuesof the trig functions arecalculated using thereference angle of the



Review

Reference Angle: theangle between a givenangle and the nearest x axis. (Note: x axis; not y

axis). Reference anglesare always positive.



Review

How you find the

reference angledepends on whichquadrant contains thegiven angle.

R i



Review

2

1

-1

-2

-2 -1 1 2 3

ReferenceAngle

GivenAngle &

R i



Review

2

1

-1

-2

-3 -2 -1 1 2

Given

Angle

Reference

Angle

R i



Review

-3 -2 -1 1 2

2

1

-1

-2

ReferenceAngle

Given

Angle

R i



Review

-3 -2 -1 1 2

2

1

-1

-2

Reference

Angle

Given

Angle

E l Give



ExampleDraw adiagramshowing the

relevantangles andsides andfind the sin,cos & tan of

56π

Given

Angle

Ref. Angle6

π

5

6

π

E l



Example

5

6

π

6

π

Short

Side

LongSide

GivenAngl

e

E l



Example

5

6

π

6

π

21

3−

Why is 1 & 2 pos &3

5 1sin

6 2

π =

5 3

cos 6 2

π = −

5 3

tan 6 3

π = −

T Thi Give



Try ThisDraw adiagramshowing the

relevantangles andsides andfind the csc,sec & cot of

56π

Given

Angle

Ref. Angle6

π

5

6

π

S l ti



Solution

5

6

π

6

π

21

3− 5csc 2

6

π =

5 2 3sec6 3π

= −

5cot 36

π = −

Find the Quadrant

al



- 1 1

- 1

1Find the

values of the six trig

functionsof the

givenangle in

standard

2

π θ

sinθ

cosθ

tanθ

cscθ

secθ

cotθ

Quadrantal

Angles

Find the Quadranta

l Angle



- 1 1

- 1

1Find the


functionsof the

givenangle in

standard

2π

sinθ

cosθ

tanθ

cscθ

secθ

cotθ

Quadrantal Angle

Find the Quadranta

l Angle



- 1 1

- 1

1Find the


functionsof the

givenangle in

standard

π

sinθ

cosθ

tanθ

cscθ

secθ

cotθ

Quadrantal Angle

Find theQuadrantal A

ngles



- 1 1

- 1

1Find the


functionsof the

givenangle in

standard

32π

sinθ

cosθ

tanθ

cscθ

secθ

cotθ

Quadrantal Angles

Important Ideas



Important Ideas

• Trig functions of quadrantal angles have

exact values.

• Trig functions of allother angles have

approximate values

• Trig functions of specialangles have exact values.



Example

Use a calculator toapproximate cos to4 decimal places.

You did not forget tocheck “mode”, did

4

9

π

−



Example

Use a calculator toapproximatecsc(11.55π ) to 4

decimal places.Check “mode” everytime.



Lesson Close

You will frequently usethe trig knowledge youlearned in Precalculusin Calculus

Lecture 1.41 Value of Trig Functions

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