Topic 11Topic 11

Periodic and Exponential Periodic and Exponential Functions IFunctions I

 

Recall

•A S T C

sinsinθθ= = = + = = = +

coscosθθ= = = += = = +

tantanθθ= = = = = = ++

X

Y

-1 1

-1

1

0

opp

hyp

+

+

adj

hyp

+

+

opp

adj

+

+

θ

sinsinθθ= = = + = = = +

coscosθθ= = = -= = = -

tantanθθ= = = - = = = - X

Y

-1 1

-1

1

0

opp

hyp

+

+

adj

hyp

-

+

opp

adj

+

-

θ

sinsinθθ= = = - = = = -

coscosθθ= = = -= = = -

tantanθθ= = = = = = ++

X

Y

-1 1

-1

1

0

opp

hyp

-

+

adj

hyp

-

+

opp

adj

-

-

θ

sinsinθθ= = = - = = = -

coscosθθ= = = += = = +

tantanθθ= = = - = = = - X

Y

-1 1

-1

1

0

opp

hyp

-

+

adj

hyp

+

+

opp

adj

-

+

θ

X

Y

-1 1

-1

1

0

ALL +

COS +TAN +

SIN +

A S T Cll entralotations

X

Y

-1 1

-1

1

0

ALL +

COS +TAN +

SIN +

A S T Cll lasseseachaints

X

Y

-1 1

-1

1

0

ALL +

COS +TAN +

SIN +

A S T Cll onstantlyalkheilas

Blah blah blah !!!!

Yappity yappity…

Sine of any angle

Cosine of any angle

Tangent of any angle

 

Recall

•A S T C

•tan = sin / cos

•sin (90-) = cos

•cos (90-) = sin

•sin2 + cos2 =1

•sin, cos and tan of 30o , 60o and 45o

sin 30o = ½

cos 30o =

tan 30o =

23

31

sin 60o =

cos 60o = ½

tan 60o =

23

13

30

60

sin 45o =

cos 45o =

tan 45o = 1

21

21

22

22

45

45

θsinθcos

θtan1

θcot

θcos1

θsec

θsin1

θcosec

θcosθsin

θtan

cosθθcos

sinθθsin

So we can express every trig ratio in terms of sin and cos

Model: Evaluate (a) cosec Model: Evaluate (a) cosec /4/4 (b) cot 5 (b) cot 5/3/3

31

32

21

23

21

3

3

35

21

sin

cos

cot)(

2

1

)(

b

sin

1coseca

4π4

π

Model: If sec Model: If sec = 5/4, find cot = 5/4, find cot if if is acuteis acute

34

43

cot

tan

cos

)(

54

45seca

3

45

Model: Prove(a) cosec2x = sec x cosec x cot x

(b) 1/(cosec x + 1) + 1/(cosec x -1) = 2tan x sec x

LHE

xcosec

xxecxsecRHEa

x

xx

xxcos1

2

sin1

sincos

sin1

2

cotcos)(

Model: Prove(a) cosec2x = sec x cosec x cot x

(b) 1/(cosec x + 1) + 1/(cosec x -1) = 2tan x sec x