Trig Cheat Sheet - Lamar University

Trig Cheat Sheet

Definition of the Trig Functions

Right triangle definition

Unit Circle Definition

For this definition we assume that

¦Ð

0 < ¦È < or 0? < ¦È < 90? .

2

For this definition ¦È is any angle.

opposite

hypotenuse

adjacent

cos(¦È) =

hypotenuse

opposite

tan(¦È) =

adjacent

sin(¦È) =

hypotenuse

opposite

hypotenuse

sec(¦È) =

adjacent

adjacent

cot(¦È) =

opposite

csc(¦È) =

y

=y

1

x

cos(¦È) = = x

1

y

tan(¦È) =

x

sin(¦È) =

1

y

1

sec(¦È) =

x

x

cot(¦È) =

y

csc(¦È) =

Facts and Properties

Domain

The domain is all the values of ¦È that can be

plugged into the function.

sin(¦È), ¦È can be any angle

cos(¦È), ¦È can be any angle





1

tan(¦È), ¦È 6= n +

¦Ð, n = 0, ¡À1, ¡À2, . . .

2

csc(¦È), ¦È 6= n¦Ð, n = 0, ¡À1, ¡À2, . . .





1

sec(¦È), ¦È 6= n +

¦Ð, n = 0, ¡À1, ¡À2, . . .

2

cot(¦È), ¦È 6= n¦Ð, n = 0, ¡À1, ¡À2, . . .

Period

The period of a function is the number, T , such

that f (¦È + T ) = f (¦È). So, if ¦Ø is a fixed number

and ¦È is any angle we have the following

periods.

2¦Ð

sin (¦Ø ¦È)

¡ú

T =

¦Ø

2¦Ð

cos (¦Ø ¦È)

¡ú

T =

¦Ø

¦Ð

tan (¦Ø ¦È)

¡ú

T =

¦Ø

2¦Ð

csc (¦Ø ¦È)

¡ú

T =

¦Ø

2¦Ð

sec (¦Ø ¦È)

¡ú

T =

¦Ø

¦Ð

cot (¦Ø ¦È)

¡ú

T =

¦Ø

Range

The range is all possible values to get out of the function.

?1 ¡Ü sin(¦È) ¡Ü 1

?1 ¡Ü cos(¦È) ¡Ü 1

?¡Þ < tan(¦È) < ¡Þ

?¡Þ < cot(¦È) < ¡Þ

sec(¦È) ¡Ý 1 and sec(¦È) ¡Ü ?1

csc(¦È) ¡Ý 1 and csc(¦È) ¡Ü ?1

? Paul Dawkins -

Trig Cheat Sheet

Formulas and Identities

Tangent and Cotangent Identities

sin(¦È)

cos(¦È)

tan(¦È) =

cot(¦È) =

cos(¦È)

sin(¦È)

Reciprocal Identities

1

csc(¦È) =

sin(¦È)

1

sec(¦È) =

cos(¦È)

1

cot(¦È) =

tan(¦È)

1

csc(¦È)

1

cos(¦È) =

sec(¦È)

1

tan(¦È) =

cot(¦È)

sin(¦È) =

Half Angle Formulas

r

 

¦È

1 ? cos(¦È)

sin

=¡À

2

2

r

 

¦È

1 + cos(¦È)

cos

=¡À

2

2

s

 

¦È

1 ? cos(¦È)

tan

=¡À

2

1 + cos(¦È)

Half Angle Formulas (alternate form)

sin2 (¦È) =

Pythagorean Identities

1

2 (1 ? cos(2¦È))

= 12 (1 + cos(2¦È))

tan2 (¦È) =

sin2 (¦È) + cos2 (¦È) = 1

cos2 (¦È)

tan2 (¦È) + 1 = sec2 (¦È)

Sum and Difference Formulas

2

1 + cot (¦È) = csc2 (¦È)

1 ? cos(2¦È)

1 + cos(2¦È)

sin(¦Á ¡À ¦Â) = sin(¦Á) cos(¦Â) ¡À cos(¦Á) sin(¦Â)

cos(¦Á ¡À ¦Â) = cos(¦Á) cos(¦Â) ? sin(¦Á) sin(¦Â)

Even/Odd Formulas

sin(?¦È) = ? sin(¦È)

csc(?¦È) = ? csc(¦È)

cos(?¦È) = cos(¦È)

sec(?¦È) = sec(¦È)

tan(?¦È) = ? tan(¦È)

cot(?¦È) = ? cot(¦È)

tan(¦Á ¡À ¦Â) =

tan(¦Á) ¡À tan(¦Â)

1 ? tan(¦Á) tan(¦Â)

Product to Sum Formulas

1

2

Periodic Formulas

sin(¦Á) sin(¦Â) =

If n is an integer then,

cos(¦Á) cos(¦Â) =

sin(¦È + 2¦Ðn) = sin(¦È) csc(¦È + 2¦Ðn) = csc(¦È) sin(¦Á) cos(¦Â) =

cos(¦È + 2¦Ðn) = cos(¦È) sec(¦È + 2¦Ðn) = sec(¦È) cos(¦Á) sin(¦Â) =

tan(¦È + ¦Ðn) = tan(¦È)

[cos(¦Á ? ¦Â) ? cos(¦Á + ¦Â)]

1

2 [cos(¦Á ? ¦Â) + cos(¦Á + ¦Â)]

1

2 [sin(¦Á + ¦Â) + sin(¦Á ? ¦Â)]

1

2 [sin(¦Á + ¦Â) ? sin(¦Á ? ¦Â)]

cot(¦È + ¦Ðn) = cot(¦È)

Sum to Product Formulas









¦Á+¦Â

¦Á?¦Â

sin(¦Á) + sin(¦Â) = 2 sin

cos

Degrees to Radians Formulas

2

2









If x is an angle in degrees and t is an angle in

¦Á+¦Â

¦Á?¦Â

sin(¦Á) ? sin(¦Â) = 2 cos

sin

radians then

2

2

¦Ð

t

¦Ðx

180t









=

?

t=

and

x=

¦Á+¦Â

¦Á?¦Â

180

x

180

¦Ð

cos(¦Á) + cos(¦Â) = 2 cos

cos

2

2









Double Angle Formulas

¦Á+¦Â

¦Á?¦Â

cos(¦Á)?cos(¦Â) = ?2 sin

sin

sin(2¦È) = 2 sin(¦È) cos(¦È)

2

2

cos(2¦È) = cos2 (¦È) ? sin2 (¦È)

2

= 2 cos (¦È) ? 1

= 1 ? 2 sin2 (¦È)

tan(2¦È) =

2 tan(¦È)

1 ? tan2 (¦È)

Cofunction Formulas

¦Ð



sin

? ¦È = cos(¦È)

2¦Ð



csc

? ¦È = sec(¦È)

 ¦Ð2



tan

? ¦È = cot(¦È)

2

¦Ð



? ¦È = sin(¦È)

2

¦Ð



sec

? ¦È = csc(¦È)

 ¦Ð2



cot

? ¦È = tan(¦È)

2

cos

? Paul Dawkins -

Trig Cheat Sheet

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

Example



cos

5¦Ð

3



1

=

2



sin

5¦Ð

3

¡Ì



=?

3

2

? Paul Dawkins -

Trig Cheat Sheet

Inverse Trig Functions

Definition

y = tan?1 (x) is equivalent to x = tan(y)

Inverse Properties




cos cos?1 (x) = x




sin sin?1 (x) = x




tan tan?1 (x) = x

Domain and Range

Alternate Notation

?1

y = sin

(x) is equivalent to x = sin(y)

y = cos?1 (x) is equivalent to x = cos(y)

Function

Domain

y = sin?1 (x)

?1 ¡Ü x ¡Ü 1

y = cos?1 (x)

?1 ¡Ü x ¡Ü 1

y = tan?1 (x)

?¡Þ < x < ¡Þ

Range

¦Ð

¦Ð

? ¡Üy¡Ü

2

2

0¡Üy¡Ü¦Ð

¦Ð

¦Ð

? ................
................

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