UNIT CIRCLE - MATHEMATICS- Basic understanding.pdf

Unit circle;
Basic mathematics;
Ekam Vartud;
Ekam circle;
Quadrant system in maths;
Quadrant understanding;
Unit circle 0 all information;

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Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

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Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

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Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

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− Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

6. Trigonometric Identities 1
Trigonometric Identities
Basic Identities:
sin =
y
r
( 1 )
cos =
x
r
( 2 )
tan =
y
x
=
sin
cos
( 3 )
cot
x
y
=
cos
sin
=
1
tan
( 4 )
sec =
r
x
=
1
cos
( 5 ) csc  =
r
y
=
1
sin
( 6 )
x
y
r

4/3
5/4
7/6
2/3
5/3



5/6
3/4



/6, 30
/4, 45
7/4
3/2








cos
sin
1
Unit Circle
0
/3, 60
11/6
/2, 90 Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

7. 2 Trigonometric Identities
Table of Values from 0 to 2:
degrees 0 30 45 60 90 120 135 150 180
radians 0 /6 /4 /3 /2 2/3 3/4 5/6 
sin
(y/r) 0
1
2
2
2
3
2
1
3
2
2
2
1
2 0
cos
(x/r) 1
3
2
2
2
1
2 0
-
1
2 -
2
2
-
3
2
-1
tan
(y/x) 0
3
3
1
3  -3
-1 -
3
3
0
cot
(x/y)
3
1
3
3
0 -
3
3
-1
-3
sec
(r/x) 1
23
3 2
2

-2
-2
-23
3
-1
csc
(r/y)

2
2
23
3
1
23
3 2
2

Table continued
degrees 210 225 240 270 300 315 330 360
radians 7/6 5/4 4/3 3/2 5/3 7/4 11/6 2
sin
(y/r)
-
1
2
-2
2
-3
2
-1
-3
2
-2
2
-
1
2 0
cos
(x/r)
-3
2
-2
2
-
1
2 0
1
2
2
2
3
2
1
tan
(y/x)
3
3
1
3 -3
-1
-3
3
0
cot
(x/y)
3
1
3
3
0
-3
3
-1
-3
sec
(r/x)
-23
3 -2
-2

2
2
23
3
1
csc
(r/y) -2
-2
-23
3
-1 -
-23
3
-2
-2
 Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

8. Trigonometric Identities 3
Some useful trigonometric relationships:
sin
2
 + cos
2
= 1 ( 7 )
1 + tan
2
= sec
2
(divide [ 7 ] by sin
2
) 
1 + cot
2
 = csc
2
(divide [ 7 ] by cos
2
) 
sin(±) = sin cos ± cos sin

cos () = coscos+ sinsin
( 11 )
tan ()=
tantan
1+ tantan
( 12 )
sin
2
=
1
2
(1 - cos 2)
( 13 )
cos
2
=
1
2
(1 + cos 2)
( 14 )
sin 2 = 2sin cos 
cos 2q = cos
2
q - sin
2
q = 2cos
2
q - 1 = 1 - 2sin
2
q ( 16 )
tan 2=
2 tan
1 - tan
2

( 17 )
sin

2
=
1 - cos
2
( 18 )
cos

2
=
1 + cos
2
( 19 ) Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

9. 4 Trigonometric Identities
tan

2
=
1 - cos
sin
=
sin
1 + cos
( 20 )
All of the above relationships are easily proved from Euler's identity
e
i
= cos  + isin , ( 21a )
and it also follows that
e
-i
= cos  - isin , ( 21b )
cos =
e
i
+ e
-i
2
= cos (-)
( 22 )
sin =
e
i
- e
-i
2i
= - sin (-) ( 23 )
and these identities can be manipulated to get a new and sometimes more
convenient expression for the trigonometric function of an angle. Just in case
you doubt this method, we append some derivations:
cos
2
+ sin
2
=
e
i
+e
-i
2
2
+
e
i
-e
-i
2i
2
=
e
2i
+2+e
-2i
4
+
e
2i
-2+e
-2i
-4
( 24 )
=
e
2i
+2+e
-2i
-e
2i
+2 - e
-2i
4
= 1 .
sin 2=
e
2i
-e
-2i
2i Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

10. Trigonometric Identities 5
=
e
i
2
-e
-i
2
2i
=
(e
i
+e
-i
)(e
i
-e
-i
)
2i
( 25 )
=(e
i
+e
-i
) sin
= 2 cos  sin  .
The hyperbolic functions are analogous to the trig functions and often arise in
physical situations. Their relations to the trig functions are as follows:
sinh x =
e
x
- e
-x
2
=
e
-i(ix)
- e
i(ix)
2
= -i sin (ix) , ( 26 )
cosh x =
e
x
+ e
-x
2
=
e
-i(ix
) + e
i(ix)
2
= cos (ix) , ( 27 )
cos x = cosh
x
i
= cosh (-ix) = cosh (ix) , ( 28 )
sin x = i sinh
x
i
= i sinh (-ix) = -i sinh (ix). ( 29 )
(The following laws apply for all
triangles with angles, A, B and C and
opposite side lengths as defined in
the figure.)
Law of Sines:
sin A
a
=
sin B
b
=
sin C
c
( 30 )
Law of Cosines: c
2
= a
2
+ b
2
- 2abcos C ( 31 )
a
b
c
B
CA Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

11. Saved C: Trigonometry Formulas {Web Page} microsoftword & PDF
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1
Trigonometric Identities & Formulas
Tutorial Services – Mission del Paso Campus

Reciprocal Identities Ratio or Quotient Identities
sin
cscx
x
1

csc
sinx
x
1

tan
sin
cosx
x
x
cot
cos
sinx
x
x


cos
secx
x
1

sec
cosx
x
1
sin x = cosx tanx cosx = sinx cotx


tan
cotx
x
1

cot
tanx
x
1

Pythagorean Identities Pythagorean Identities in Radical Form

sin cos
22
1
x x sin cosxx 1
2


1
22
tan sec
x x

1
22
cot csc
x x tan secxx 
2
1

Note: there are only three, basic Pythagorean identities, the other forms
cos sinxx 1
2

are the same three identities, just arranged in a different order
.
Confunction Identities Odd-Even Identities
Also called negative angle identities
sin cos

2






xx cos sin

2






xx Sin (- x) = -sin x Csc (-x) = -csc x
Cos (- x) = cos x Sec (-x) = sec x
tan cot

2






xx cot tan

2






xx Tan (- x) = -tan x Cot (-x) = -cot x

sec csc

2






xx csc sec

2






xx Phase Shift =
c
b


Period =
2
b


Sum and Difference Formulas/Identities How to Find Reference Angles
sin( ) sin cos cos sinuv u v u v  Step 1: Determine which quadrant the angle is in
sin( ) sin cos cos sinuv u v u v  Step 2: Use the appropriate formula
Quad I = is the angle itself
cos( ) cos cos sin sinuv u v u v  Quad II = 180 – θ or π - θ
cos( ) cos cos sin sinuv u v u v  Quad III = θ – 180 or θ - π
Quad IV = 360 – θ or 2π - θ
tan( )
tan tan
tan tanuv
uv
uv


1

tan( )
tan tan
tan tanuv
uv
uv


1
Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

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2


Reciprocal Identities Ratio or Quotient Identities
sin
cscx
x
1

csc
sinx
x
1

tan
sin
cosx
x
x
cot
cos
sinx
x
x



cos
secx
x
1

sec
cosx
x
1
sin x = cosx tanx cosx = sinx cotx



tan
cotx
x
1

cot
tanx
x
1


Pythagorean Identities Pythagorean Identities in Radical Form
sin cos
22
1x x sin cosxx 1
2


1
22
tan sec
x x

1
22
cot csc
x x tan secxx 
2
1

Note: there are only three, basic Pythagorean identities, the other forms
are the same three identities, just arranged in a different order
.


Confunction Identities Odd-Even Identities
Also called negative angle identities
sin cos

2






xx cos sin

2






xx Sin (- x) = -sin x Csc (-x) = -csc x
Cos (- x) = cos x Sec (-x) = sec x
tan cot

2






xx cot tan

2






xx Tan (- x) = -tan x Cot (-x) = -cot x

sec csc

2






xx csc sec

2






xx


Sum and Difference Formulas - Identities

sin( ) sin cos cos sinuv u v u v  cos( ) cos cos sin sinuv u v u v 
sin( ) sin cos cos sinuv u v u v  cos( ) cos cos sin sinuv u v u v 


tan( )
tan tan
tan tanuv
uv
uv


1

tan( )
tan tan
tan tanuv
uv
uv


1





Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

13. Saved C: Trigonometry Formulas {Web Page} microsoftword & PDF
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3


The Unit Circle
90°

Tan = -
3 cot 
3
3
tan = undefined & cot= 0 tan =
3 cot =
3
3


120° 60°

Tan = 1
Tan =- - 1 cot = 1
Cot = -1
135° 45°



2.09 1.57
1.04
150° 2.35 30°

.785
2.61
Tan =

3
3
cot = -
3 .523 tan =
3
3
cot =
3


3.14
Tan= 0 Tan=0 & cot=undef
Cot=undef 180° 360°

3.66 2(3.14 )= 6.28

Tan
3
3
cot =
3 3.925 5.75 tan = 
3
3
cot = -
3

4.186 5.49

4.71 5.23 330°
210°



Tan = -1
Tan = 1 Cot = -1
Cot = 1

225° 315°




240° 270° 300°
Tan = 3 cot =
3
3
tan=undefined tan = -
3 cot = 
3
3

Cot = 0


Definition of Trigonometric Functions concerning the Unit Circle

sin θ =
opp
hyp
y
r
 csc θ =
hyp
opp
r
y


cos θ =
adj
hyp
x
r
 sec θ =
hyp
adj
r
x



tan θ =
opp
adj
y
x
 cot θ =
adj
opp
x
y

Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

14. Saved C: Trigonometry Formulas {Web Page} microsoftword & PDF
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4


Right Triangle Definitions of Trigonometric Functions
Note: sin & cos are complementary angles, so are tan & cot and sec & cos, and the sum of complementary angles is 90 degrees.
C
sin θ =
opp
hyp
y
r
 csc θ =
hyp
opp
r
y

r y

Hypotenuse opposite
cos θ =
adj
hyp
x
r
 sec θ =
hyp
adj
r
x

A
x B
adjacent
tan θ =
opp
adj
y
x
 cot θ =
adj
opp
x
y


Adjacent =
is the side adjacent to the angle in consideration. So if we are considering Angle A, then the adjacent side is CB




Trigonometric Values of Special Angles
Degrees 0° 30° 45° 60° 90° 180° 270°

Radians

0



6


4


3




2






3
2



sinθ

0

1
2



2
2

3
2


1

0

-1

cosθ

1

3
2



2
2


1
2


0

-1

0

tanθ

0

3
3



1


3

undefined

0

undefined


To Convert Degrees to Radians, Multiply by
 rad
180deg


To Convert Radians to Degrees, Multiply by
180deg
 rad




Vocabulary
 Cotangent Angles - are two angles with the same terminal side
 Reference Angle - is an acute angle formed by terminal side of angle(α) with x-axis

Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

15. Saved C: Trigonometry Formulas {Web Page} microsoftword & PDF
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5

Double Angle Identities Half Angle Identities Power Reducing Formulas
sin sin cos22A AA sin
cos
AA
2
1
2


sin
cos
2
12
2
u
u


cos cos sin2
22
AAA
cos
cos
AA
2
1
2


cos
cos
2
12
2
u
u


cos cos22 1
2
AA
tan
cos
sin
AA
A
2
1


tan
cos
cos
2
12
12
u
u
u


cos si
n212
2
AA

tan
tan
tan
2
2
1
2A
A
A

tan
sin
cos
AA
A
21




Product-to-Sum Formulas Sum-to-Product Formulas
sin sin cos( ) cos( )uv uv uv
1
2

sin sin sin cosxy
xy xy











 2
22

cos cos cos( ) cos( )uv uv uv
1
2

sin sin cos sinxy
xy xy











 2
22

sin cos sin( ) sin( )uv uv uv
1
2

cos cos cos cosxy
xy xy











 2
22

cos sin sin( ) sin( )uv uv uv
1
2

cos cos sin sinxy
xy xy











 2
22

Law of Sines Law of Cosines
Solving Oblique Triangles using sine: AAS, ASA, SSA, SSS, SAS Cosine: SAS, SSS

Standard Form Alternative Form
a
A
b
B
c
C
sin sin sin
 or
sin sin sinA
a
B
b
C
c
 abc bc A
222
2
cos cosA
bca
bc

22 2
2


bac ac
B
222
2cos cosB
acb
ac

222
2


cba abC
222
2
cos cosC
abc
ab

222
2


Finding the Area of non-90degree Triangles

Area of an Oblique Triangle Heron’s Formula

area bc A ab C ac B
1
2
1
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1
2
sin sin sin
Step 1: Find “s”
s
abc

2

Step 2: Use the formula
area s s a s b s c()()() Unit Circle and Trigonometry formulas
Prof. S. P. Parmar

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