!Trig Cheat Sheet - UCI Mathematics

[Pages:2]Trig Cheat Sheet

Definition of the Trig Functions

Right triangle definition

For this definition we assume that

Unit circle definition

0 < < or 0? < < 90? . 2

For this definition is any angle.

y

( x, y)

opposite

hypotenuse

adjacent

1 y

x x

sin = opposite hypotenuse

cos = adjacent hypotenuse

tan = opposite adjacent

csc = hypotenuse opposite

sec = hypotenuse adjacent

cot = adjacent opposite

sin = y = y 1

cos = x = x 1

tan = y x

csc = 1 y

sec = 1 x

cot = x y

Facts and Properties

Domain

The domain is all the values of that

Period

can be plugged into the function.

The period of a function is the number,

sin , cos ,

tan ,

csc ,

can be any angle can be any angle

n

+

1 2

,

n = 0, ?1, ? 2,...

n , n = 0, ?1, ? 2,...

T, such that f ( + T ) = f ( ) . So, if

is a fixed number and is any angle we have the following periods.

sin ( ) T = 2

sec ,

n

+

1 2

,

n = 0, ?1, ? 2,...

cot , n , n = 0, ?1, ? 2,...

cos ( ) tan ( )

T

=

2

T =

Range The range is all possible values to get

csc( ) T = 2

out of the function.

-1 sin 1 csc 1 and csc -1

-1 cos 1 sec 1 and sec -1

- tan

- cot

sec ( ) cot ( )

T = 2

T =

? 2005 Paul Dawkins

Formulas and Identities

Tangent and Cotangent Identities

Half Angle Formulas

tan = sin cos

cot = cos sin

sin2 = 1 (1- cos (2 ))

2

Reciprocal Identities

csc = 1 sin

sin = 1 csc

sec = 1 cos

cos = 1 sec

cot

=

1 tan

tan

=

1 cot

Pythagorean Identities

cos2 = 1 (1+ cos(2 ))

2

tan 2

=

1- 1+

cos (2 cos ( 2

) )

Sum and Difference Formulas

sin ( ? ) = sin cos ? cos sin

cos ( ? ) = cos cos sin sin

sin2 + cos2 = 1 tan2 +1 = sec2 1+ cot2 = csc2

Even/Odd Formulas

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

cos (- ) = cos

sec(- ) = sec

tan

(

?

)

=

tan ? tan 1 tan tan

Product to Sum Formulas

sin

sin

=

1 2

cos (

-

)

-

cos (

+

)

cos

cos

=

1 2

cos (

-

)

+

cos (

+

)

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

Periodic Formulas If n is an integer.

sin ( + 2 n) = sin csc( + 2 n) = csc

sin

cos

=

1 2

sin (

+

)

+ sin (

-

)

cos

sin

=

1 2

sin (

+

)

- sin

(

-

)

Sum to Product Formulas

cos ( + 2 n) = cos sec( + 2 n) = sec tan ( + n) = tan cot ( + n) = cot

Double Angle Formulas

sin (2 ) = 2sin cos cos (2 ) = cos2 - sin2

= 2 cos2 -1 = 1- 2 sin2

sin

+

sin

=

2 sin

+ 2

cos

- 2

sin

- sin

=

2

cos

+ 2

sin

- 2

cos

+

cos

=

2

cos

+ 2

cos

- 2

cos

-

cos

=

-2 sin

+ 2

sin

- 2

tan

(

2

)

=

1

2 -

tan tan2

Degrees to Radians Formulas

Cofunction Formulas

sin

2

-

=

cos

cos

2

-

=

sin

If x is an angle in degrees and t is an

angle in radians then

= t t = x and x = 180t

180 x

180

csc

2

-

=

sec

tan

2

-

=

cot

sec

2

-

=

csc

cot

2

-

=

tan

? 2005 Paul Dawkins

Unit Circle

-

1,

3

2 2

-

2, 2

2

2

3

2 3 120?

-

3

,

1

2 2

5 4 135?

6 150?

y (0,1)

2

90?

(-1,0) 180?

1, 2

3 2

3

60?

4

45?

2, 2

2 2

3 2

,

1 2

6 30?

0?

0 (1,0)

360? 2 x

210? 7

6

-

3

,

-

1

2 2

-

2 ,-

2

5 4

225? 4

240?

2 2

3

-

1,- 2

3

2

270? 3 2

(0,-1)

330? 11

315?

300?

7

6

3 2

,-

1 2

5

4

2 ,-

2

3

2 2

1,-

3

2 2

For any ordered pair on the unit circle ( x, y) : cos = x and sin = y

Example

cos

5 3

=

1 2

sin

5 3

=

-

3 2

Inverse Trig Functions

Definition y = sin-1 x is equivalent to x = sin y

Inverse Properties

cos (cos-1 ( x)) = x

y = cos-1 x is equivalent to x = cos y y = tan-1 x is equivalent to x = tan y

sin (sin-1 ( x)) = x tan (tan-1 ( x)) = x

cos-1 (cos( )) = sin-1 (sin ( )) = tan-1 (tan ( )) =

Domain and Range Function Domain y = sin-1 x -1 x 1

y = cos-1 x -1 x 1

y = tan-1 x - < x <

Range

- y

2

2

0 y

- < y<

2

2

Alternate Notation sin-1 x = arcsin x cos-1 x = arccos x tan-1 x = arctan x

Law of Sines, Cosines and Tangents

c

a

Law of Sines

sin = sin = sin

a

b

c

Law of Cosines a2 = b2 + c2 - 2bc cos b2 = a2 + c2 - 2ac cos c2 = a2 + b2 - 2ab cos

Mollweide's Formula

a

+

b

=

cos

1 2

(

-

)

c

sin

1 2

b

Law of Tangents

a a

- +

b b

=

tan tan

1 2

1 2

( (

- +

) )

b b

- +

c c

=

tan tan

1 2

1 2

( (

- +

) )

a a

- +

c c

=

tan tan

1 2

1 2

( (

- +

) )

? 2005 Paul Dawkins

? 2005 Paul Dawkins

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