Trig Cheat Sheet - Lamar University

Trig Cheat Sheet

Definition of the Trig Functions

Right triangle definition

For this definition we assume that 0 < < or 0 < < 90.

2

Unit Circle Definition For this definition is any angle.

opposite sin() = hypotenuse

adjacent cos() =

hypotenuse tan() = opposite

adjacent

hypotenuse csc() = opposite

hypotenuse sec() =

adjacent cot() = adjacent

opposite

sin() = y = y 1

cos() = x = x 1

tan() = y x

Facts and Properties

csc() = 1 y

sec() = 1 x

cot() = x y

Domain

Period

The domain is all the values of that can be plugged into the function.

The period of a function is the number, T , such that f ( + T ) = f (). So, if is a fixed number

sin(), can be any angle

cos(), can be any angle

tan(), =

1 n+

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

2

csc(), = n, n = 0, ?1, ?2, . . .

sec(), =

1 n+

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

2

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

Range

and is any angle we have the following

periods.

2 sin ( ) T =

2 cos ( ) T =

tan ( ) T =

2 csc ( ) T =

2 sec ( ) T =

cot ( )

T=

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

-1 sin() 1 - < tan() < sec() 1 and sec() -1

-1 cos() 1 - < cot() < csc() 1 and csc() -1

? Paul Dawkins -

Trig Cheat Sheet

Formulas and Identities

Tangent and Cotangent Identities

tan() = sin() cos()

cot() = cos() sin()

Reciprocal Identities csc() = 1 sin() sec() = 1 cos() 1 cot() = tan()

sin() = 1 csc()

cos() = 1 sec() 1

tan() = cot()

Pythagorean Identities sin2() + cos2() = 1

tan2() + 1 = sec2()

1 + cot2() = csc2()

Half Angle Formulas

sin = ? 1 - cos()

2

2

cos = ? 1 + cos()

2

2

1 - cos()

tan

=?

2

1 + cos()

Half Angle Formulas (alternate form)

sin2()

=

1 2

(1

-

cos(2))

tan2() = 1 - cos(2)

cos2()

=

1 2

(1

+

cos(2))

1 + cos(2)

Sum and Difference Formulas

sin( ? ) = sin() cos() ? cos() sin()

Even/Odd Formulas

cos( ? ) = cos() cos() sin() sin()

sin(-) = - sin() cos(-) = cos()

csc(-) = - csc() tan( ? ) = tan() ? tan()

sec(-) = sec()

1 tan() tan()

tan(-) = - tan() Periodic Formulas If n is an integer then,

cot(-) = - cot()

Product to Sum Formulas

sin() sin()

=

1 2

[cos(

-

)

-

cos(

+

)]

cos() cos()

=

1 2

[cos(

-

)

+

cos(

+

)]

sin( + 2n) = sin()

csc( + 2n) = csc()

sin() cos() =

1 2

[sin(

+

)

+

sin(

-

)]

cos(

+ 2n)

=

cos()

sec(

+ 2n)

=

sec()

cos() sin()

=

1 2

[sin(

+ ) - sin( - )]

tan( + n) = tan() cot( + n) = cot()

Degrees to Radians Formulas

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

radians then

t

x

180t

=

t=

and x =

180 x

180

Double Angle Formulas sin(2) = 2 sin() cos()

Sum to Product Formulas

sin() + sin() = 2 sin + cos -

2

2

sin() - sin() = 2 cos + sin -

2

2

cos() + cos() = 2 cos + cos -

2

2

cos()-cos() = -2 sin + sin -

2

2

cos(2) = cos2() - sin2() = 2 cos2() - 1 = 1 - 2 sin2() 2 tan()

tan(2) = 1 - tan2()

Cofunction Formulas

sin

-

= cos()

2

csc - = sec()

2

tan

-

= cot()

2

cos

-

= sin()

2

sec - = csc()

2

cot

-

= tan()

2

? Paul Dawkins -

Trig Cheat Sheet

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

Example

5 1

cos

=

32

5

3

sin

=-

3

2

? Paul Dawkins -

Trig Cheat Sheet

Inverse Trig Functions

Definition y = sin-1(x) is equivalent to x = sin(y) y = cos-1(x) is equivalent to x = cos(y) 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

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

0y

- ................
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