3.5: Trigonometric Functions Reference Evans 6.1

Consider a right-angled triangle with angle θ and side lengths x, y and h asshown:

θx

yh

The trigonometric functions sine, cosine and tangent of θ are defined as:

sin θ =opposite

hypotenuse=

y

h, cos θ =

adjacent

hypotenuse=

x

h

tan θ =opposite

adjacent=

y

x=

sin θ

cos θ

71

Reciprocal Trigonometric Functions

The reciprocal trigonometric functions secant, cosecant and cotangent are de-fined as:

sec θ =1

cos θ

cosec θ =1

sin θ

cot θ =1

tan θor

cos θ

sin θ

72

Example

θ12

513

For the above right-angled triangle, find:

1. sin θ

2. cos θ

3. tan θ

4. sec θ

5. cosec θ

6. cot θ

You can now attempt Sheet 3 Q19

73

The Unit Circle

We can form right-angled triangles in a unit circle (circle of radius 1).

If θ is the anticlockwise angle between the positive x-axis and the ray−−→OP then for

all θ ∈ R:

sin θ =y

1= y , cos θ =

x

1= x and tan θ =

y

x

1

x

y

x

y

θ

1

1O

P(x,y)

1

x

y

x

y

θ

1

1O

P(x,y)

1

x

y

x

y

θ

1

1

negative θ(clockwise)

O

P(x,y)

74

Signs of Trigonometric Functions

The signs of the trigonometric functions for different values of θ can be determinedby noting the signs of x and y in the unit circle:

1

x

y

x

y1

1

y

x

1st quadrant x positive y positive

4th quadrant x positive y negative

2nd quadrant x negative y positive

3rd quadrant x negative y negative

75

‘CAST’ Diagram

Since in the unit circle sin θ = y, cos θ = x and tan θ = sin θ

cos θ = y

x, we obtain the

following ‘CAST’ diagram as a shorthand for remembering signs of trigonometricfunctions in different parts of the circle:

x

y1

1

ALL sin, cos, tan positive

only COS positive

only SIN positive

only TAN positive

C

AS

T

76

Degrees and Radians

In the unit circle, if we measure the length of the arc from A to B in an anticlock-wise direction, we have another way of measuring angles.

A

B

1

1

.

.θ

This length is called the radian measure of the angle θ and we can signify it bythe symbol θc rather than θo, but usually we don’t show units for radians.

77

Radians on the Unit Circle

Since the circumference of the unit circle is 2πr = 2π · 1 = 2π, a completeangle around the whole circle is 2π radians. Hence if we divide the circle into fourquarters or quadrants we have radian measures as shown:

0, 2ππ

π2

3π2

78

Converting Between Degrees and Radians

To convert from degrees to radians multiply by π180.

To convert from radians to degrees multiply by 180π

.

Examples

Convert 360◦,1◦,90◦,135◦ to radians.

Convert π,1, π3,7π6 radians to degrees.

79

Two Important Triangles

The following triangles help us work out the exact values of sin, cos and tan ofcertain special angles.

π

π π

π3 3

6 6

1 1

2 23

2

1

1

π

π4

4

80

Exact Values using Triangles

Using the two important triangles on the previous slide and your knowledge of theunit circle complete the table with exact values:

θ sin θ cos θ tan θ

0π

6

π

4

π

3

π

2

π

3π2

***Important to know these ratios, either the triangles or the table

81

Examples: Find the exact values of the following trig ratios:

1. cos�5π6

�

2. sin�4π3

�Ask yourself:

• quadrant?

• angle from x-axis?

• ± CAST?

• value of ratio?

3. tan�3π4

�

4. sin�11π6

�

5. tan�−π

4

�

6. cos�−2π3

�

82

Finding Angles

Here we need to work backwards considering both the value of the ratio and itssign, as this determines the quadrants of the resultant angles.Find all values of θ between 0 and 2π satisfying:

1. sin(θ) = 12

2. tan(θ) = 1

3. cos(θ) = − 1√2

83

Solving Trigonometric Equations

Solve the following trigonometric equations for x ∈ [0,2π).

1. cosx+ 12 = 0

2. 2 sin2 x+ sinx− 1 = 0

84

3. cosx+ sinx = 0

4. 2 tan2 x− 6 = 0

You can now attempt Sheet 3 Q20-21

85

Sine and Cosine Graphs Reference Evans 6.2

Plotting the values of f(x) = sinx and g(x) = cosx for x ∈ R gives thefollowing graphs

�2 Π �Π Π 2 Π 3 Π 4 Πx

�1

1f �x�

�2 Π �Π Π 2 Π 3 Π 4 Πx

�1

1g�x�

86

Properties of Sine and Cosine Graphs

• The sine and cosine functions repeat themselves after an interval (or period) of2π units. That is:

. . . sin(x− 2π) = sinx = sin(x+2π) = sin(x+4π) . . ., and

. . . cos(x− 4π) = cos(x− 2π) = cosx = cos(x+2π) . . ., for all x ∈ R

These functions are therefore said to be periodic or cyclic, with period 2π.

• The maximum and minimum values of sinx and cosx are 1 and −1 respec-tively. The graphs of f(x) = sinx and g(x) = cosx are therefore said to havean amplitude of 1.

87

Graph of Tangent Function Reference Evans 6.6

The function h(x) = tanx = sinx

cosx is defined for x ∈ R such that cosx �= 0.

Plotting h(x) = tanx over its domain x ∈ R\{(2k + 1)π2 | k ∈ Z} gives thefollowing graph

 

88

Properties of Tan Graph

• The tangent function h(x) = tanx is undefined at x = {(2k +1)π2 | k ∈ Z}(this is where cosx = 0).

The graph has vertical asymptotes at these x-values, which are usually indicatedby dotted or dashed vertical lines.

• The tangent function repeats itself after an interval of π units. That is:

tan θ = tan(θ + π), for all θ ∈ R\{(2k +1)π

2| k ∈ Z}

This function therefore has period π.

89

Dilations of Trig Graphs Reference Evans 6.2, 6.3 & 6.6

Sketch the following graphs - the standard sin or cos graph is shown - state theperiod and amplitude of the transformed graph in each case.Note: a dilation in the x direction (e.g. y = cos bx) changes the period

�to 2π

b

�

and one in the y direction (e.g. y = a sinx) changes the amplitude (to a)

 

 

1. y = sin2x 2. y = 3cosx3. y = −5

2 sinx 4. y = cos�x

2

�

90

Translations of Trigonometric Graphs

Sketch the following graphs on the axes below –adding or subtracting a value tothe trig function will translate the graph up or down, while adding or subtracting avalue to the x term will translate the graph to the left or right respectively.

1. y = tan(x− π

4)

 

91

2. y = sin(x) + 3

 

3. y = cos(x+ π)

 

92

Multiple TransformationsSketch the following sequence of graphs on the same set of axes.y1 = sinx y2 = −2 sinx y3 = −2 sin 3x y4 = −2 sin

�3x− π

2

�

 

You can now attempt Sheet 3 Q22-23

93

DE

PA

RT

ME

NT

OF

MA

TH

EM

AT

ICS

AN

DS

TA

TIS

TIC

S

MAST10012In

troduction

toM

ath

ematics

Semester1,2011

REVIS

ION

-TRIG

ONOM

ETRY

A:

Findingtrig

ratiosin

theUnit

Circle

1.Id

enti

fyth

equad

rant

that

the

angl

eis

in:

•Q

1has

angl

esfr

om0→

π 2

•Q

2has

angl

esfr

omπ 2→

π

•Q

3has

angl

esfr

omπ→

3π 2

•Q

4has

angl

esfr

om3π 2→

2π

Q1

Q2

Q3

Q4

Thi

sis

just

the

firs

tre

volu

tion

ofth

eu

nit

circ

leW

eca

nof

cou

rse

fin

dbi

gger

angl

esby

mov

ing

arou

nd

the

circ

lem

ore

than

once

orn

egat

ive

angl

esby

goin

gin

the

oppo

site

dire

ctio

n

2.D

ecid

eif

the

rati

oyo

unee

dto

find

(usu

ally

sin,

cos

orta

n)

isp

osit

ive

orneg

ativ

ein

the

quad

rant

you

found

inst

ep1

(mos

tst

ude

nts

use

CA

ST

tore

mem

ber

the

sign

s)

3.U

seth

esp

ecia

ltr

iangl

esto

find

the

rati

ore

quir

ed

2

1

3

3

6

2

1

4

4 1

(rem

embe

rS

OH

-CA

H-T

OA

)

e.g.

sinπ 6

=1 2,

cosπ 6

=√3 2,

tanπ 6

=1 √3

4.F

oran

gles

that

give

poi

nts

onth

ex

ory

axes

we

use

the

bas

icdefi

nit

ions:

•co

sθis

thexco

ord

inate

•si

nθis

theyco

ord

inate

•ta

nθ

=si

nθ

cosθ

Ther

efor

ew

eca

nfind

valu

eslike

:

cosπ

=−

1or

sin( −π 2

) =−

1or

tan

3π 2

=(s

in3π 2)

(cos

3π 2)

=−

1 0=

undefi

ned

Itm

ayhel

pto

look

atth

eunit

circ

leab

ove

tose

eth

ese

poi

nts

(and

rem

emb

erth

at−π 2

and

3π 2

are

just

diff

eren

tnam

esfo

rth

esa

me

poi

nt

onth

eci

rcle

.

1

PRACTIC

EEXERCIS

EA

1.F

ind

the

valu

esof

the

follow

ing

trig

rati

os-

the

step

sm

enti

oned

onth

epre

vio

us

pag

ehav

eb

een

spel

tou

tfo

rth

efirs

tfe

wques

tion

s(t

hen

you

are

onyo

ur

own

todo

the

rest

by

follow

ing

the

sam

epro

cess

):

(a)

sin

5π 4

i.5π 4

isin

Q..

....

(as

5π 4

=π

+π 4)

ii.

InQ

....

..si

nis

....

..(+

or–)

iii.

Fro

mtr

iangl

e2

we

know

that

sinπ 4

=..

....

=⇒

sin

5π 4

=

(b)

cos11π 6

i.11π 6

isin

Q..

....

(as

11π 6

=2π−

π 6)

ii.

InQ

....

..co

sis

....

..(+

or–)

iii.

Fro

mtr

iangl

e1

we

know

that

cosπ 6

=..

....

=⇒

cos11π 6

=

(c)

tan

8π 3

i.8π 3

isin

Q..

....

(as

8π 3

=3π−

π 3

=π−

π 3)

ii.

InQ

....

..ta

nis

....

..(+

or–)

iii.

Fro

mtr

iangl

e1

we

know

that

tanπ 3

=..

....

=⇒

tan

8π 3

=

(d)

cos

5π

i.5π

ison

the

....

..ax

is(x

ory)

ii.

Ithas

coor

din

ates

(...,...

)

iii.

As

cos

isth

ex

coor

din

ate

we

know

:

=⇒

cos

5π=

2

2.N

owtr

yth

esa

me

pro

cess

wit

hth

ese

ques

tion

s:

(a)

sin(5π 3)

(b)

cos(

5π 4)

(c)

tan(7π 6)

(d)

sin(2π

)

(e)

cos(−

7π 2)

(f)

sin(−

7π 4)

(g)

tan(−

17π 6)

Ther

ear

eot

her

trig

rati

osw

eca

nuse

but

they

are

bas

edon

the

stan

dar

don

es.

So

tofind

cose

c,se

cor

cot

we

calc

ula

tesi

n,

cos

orta

nre

spec

tive

lyan

dth

enju

st”t

urn

them

upsi

de

dow

n”

bec

ause

ofth

edefi

nit

ions:

cose

cθ

=1

sinθ

secθ

=1

cosθ

cotθ

=1

tanθ

=co

sθ

sinθ

Exam

ple

s:W

efo

und

onpag

e1

inse

ctio

n3

that

:

sinπ 6

=1 2

=⇒

cose

cπ 6

=2 1

=2

and

cosπ 6

=√3 2

=⇒

secπ 6

=2 √3

(or

2√3

3)

tanπ 6

=1 √3

=⇒

cotπ 6

=√

3

3.F

ind

the

follow

ing

trig

rati

os(y

oum

ayfind

your

answ

ers

toQ

2use

fulin

som

eca

ses)

:

(a)

cose

c(5π 3)

(b)

sec

(−7π 2)

(c)

cot(

7π 6)

(d)

sec

(3π 4)

(e)

cot(

6π)

3

B:

SolvingTrigequations

1.R

earr

ange

the

equat

ion

tom

ake

the

trig

rati

o(u

sual

lysi

n,

cos

orta

n)

the

sub

ject

e.g.

sinx

+1

=0

=⇒

sinx

=−

1√

2co

s(x

+5π 6)−

1=

0=⇒

cos(x

+5π 6)

=1 √2

2.F

ind

the

bas

ican

gle

that

sati

sfies

this

rati

o–

this

may

invo

lve

look

ing

atth

ean

gles

inth

etw

osp

ecia

ltr

ian

gles

orlo

okin

gat

the

coor

din

ates

ofpo

ints

onth

eu

nit

circ

lew

here

they

inte

rsec

tw

ith

the

two

axes

.

3.D

ecid

ew

hic

hquad

rants

the

answ

ers

must

be

in-

look

atth

esi

gnof

the

trig

rati

oe.

g.if

the

sin

rati

ohas

aneg

ativ

ean

swer

then

angl

esm

ust

be

inQ

3an

dQ

4if

the

cos

rati

ohas

ap

osit

ive

answ

erth

enan

gles

must

be

inQ

1an

dQ

4so

we

are

real

lylo

okin

gat

our

CA

ST

diag

ram

”bac

kwar

ds”

4.U

seknow

ledge

ofth

eunit

circ

leto

find

the

bas

ican

gle

inth

eri

ght

quad

rants

So

the

unit

circ

leon

the

righ

tm

ight

be

use

ful

her

e

Not

eth

atθ

isth

ebas

ican

gle

found

inst

ep2

5.C

hec

kth

edom

ain

ofth

eques

tion

haveyou

found

all

theso

lutions?

you

may

nee

dto

add

orsu

btra

ct2π

ifyo

un

eed

bigg

eran

gles

orn

egat

ive

angl

esfo

rth

edo

mai

ngi

ven

inth

equ

esti

on-

that

is,

angl

esou

tsid

eth

est

anda

rd0→

2πw

hich

ison

lyon

ere

volu

tion

arou

nd

the

un

itci

rcle

PRACTIC

EEXERCIS

EB

Sol

veth

efo

llow

ing

trig

equat

ions

over

the

give

ndom

ains

(as

inE

xer

cise

Ayo

uar

egu

ided

thro

ugh

the

firs

tfe

wques

tion

sth

enyo

ush

ould

use

the

sam

epro

cess

toco

mple

teth

ere

st):

1.2

cosx

=√

3x∈

(0,2π

)

cosx

=...

...

bas

ican

gleθ

=π 6

see

spec

ial

tria

ngl

e1

cos

is+

inQ

...(θ)

and

Q..

.(2π−θ)

=⇒

x=

π 6,

2π−

π 6

=⇒

x=

π 6,

... 6

(mentalcheck

:th

ese

valu

esar

ein

the

dom

ain

0→

2π)

Not

e:if

we

added

orsu

btr

acte

d2π

toei

ther

ofou

ran

swer

sw

ew

ould

get

valu

esou

tsid

eth

edom

ain,

sow

edon

’tnee

dto

do

anyth

ing

else

her

e

4

2.ta

n(x−

π 4)−√

3=

0x∈

(0,2π

)

tan(x−

π 4)

=...

bas

ican

gleθ

=...

see

spec

ial

tria

ngl

e1

tan

is..

.(+

/–)

inQ

...(θ)

and

Q3

(......)

=⇒

x−

π 4=......,

......

=⇒

x−

π 4=......,

......

=⇒

x=......

+π 4,

......

+π 4

add

π 4to

RH

Sof

equat

ion

tofindx

=⇒

x=......,

......

use

LC

Dto

add

frac

tion

s

(men

tal

chec

k:

thes

eva

lues

are

insi

deth

edo

mai

nof

0→

2π,

and

ifw

ead

ded

orsu

btra

cted

2πto

eith

erof

them

we

wou

ldge

tan

swer

sou

tsid

eth

edo

mai

n,

sow

edo

n’t

nee

dto

doan

ythi

ng

else

here

)

3.8

sin(x

+π 6)

+4

=0

x∈

(−2π,2π

)

sin(x

+π 6)

=...

bas

ican

gleθ

=...

see

spec

ial

tria

ngl

e1

sin

is–

inQ

3(π

+θ)

and

Q..

.(......

)=⇒

x+

π 6=......,

......

=⇒

x+

π 6=......,

......

=⇒

x=......−

π 6,

......−

π 6su

btr

actπ 6

toR

HS

ofeq

uat

ion

tofindx

=⇒

x=......,

......

use

LC

Dto

add

frac

tion

s

and

x=......−

2π,

......−

2πdom

ain

nee

ds

neg

ativ

ean

gles

too

Not

e:w

en

eed

all

fou

ran

swer

s–

the

two

neg

ativ

ean

dth

eor

igin

altw

opo

siti

veon

es

So

the

final

solu

tion

sar

e:

x=......,

......,

......,

......

4.√

2co

sx

+1

=0

x∈

(−2π,2π

)

5.si

n(x

+π 3)

=1

x∈

(0,4π

]

6.co

s(x−

3π 4)

=0

x∈

(0,2π

)

7.2

cos2x

+3

cosx

+1

=0

x∈

[0,2π

] 5

Trig

on

om

etry

Frid

ay,

22

Ap

ril 2

01

1

3:0

5 P

M

Rev

isio

n t

op

ics P

ag

e 1

Rev

isio

n t

op

ics P

ag

e 2

Rev

isio

n t

op

ics P

ag

e 3

Rev

isio

n t

op

ics P

ag

e 4

Rev

isio

n t

op

ics P

ag

e 5

Rev

isio

n t

op

ics P

ag

e 6