Trigonometry CheatSheet
1
How to use this document
This document is not meant to be a list of formulas to be learned by heart. The ﬁrst few formulas
are very basic (they descend from the deﬁnition and/or Pythagoras’ theorem) and you might want
to memorize them, but you should be able to retrieve everything else from these two/three basic
formulas, and you should be able to do it quickly (during an exam you don’t want to invest
ten minutes in recalculating a formula that you might or might not need!!!). These formulas are
arranged in a logical order, starting from the most basic, so that each formula can be retrieved
using formulas that come ﬁrst only. Every formula is accompanied by a short explanation about
how you can retrieve it (there might be more than one method).
2
Formulas that come from Pythagoras’ theorem and/or the def
inition
cos
2
x
+
sin
2
x
= 1
(1)
It’s an immediate consequence of the deﬁnition and Pythagoras’ theorem
tan
2
x
+ 1 = sec
2
x
(2)
It follows from the deﬁnition of tangent, secant and the previous formula
cot
2
x
+ 1 = csc
2
x
(3)
Same as before
cos (

x
) = cos
x
(4)
Cosine is even. It follows from the deﬁnition.
sin (

x
) =

sin
x
(5)
tan (

x
) =

tan
x
(6)
cot (

x
) =

cot
x
(7)
Sine, tangent and cotangent are all odd. It follows from the deﬁnition. The period of sine, cosine
secant and cosecant is 2
π
, and the period of tangent and cotangent is
π
:
sin (
x
+ 2
kπ
) = sin
x
(8)
cos (
x
+ 2
kπ
) = cos
x
(9)
tan (
x
+
kπ
) = tan
x
(10)
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document3
Addition of angles
You might want to take the ﬁrst two formulas as black boxes and memorize them. However, if you
know about the complex numbers you can retrieve them both very quickly.
sin (
x
+
y
) = sin
x
cos
y
+ cos
x
sin
y
(11)
cos (
x
+
y
) = cos
x
cos
y

sin
x
sin
y
(12)
These two formulas can be derived using the property of exponentials
e
i
(
x
+
y
)
=
e
ix
e
iy
Plug the Euler’s identity
e
ix
= cos
x
+
i
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 greenwood
 Calculus, Trigonometry, Formulas, Cos, lim, Inverse function, Inverse trigonometric functions

Click to edit the document details