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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