Download PDF document!

Brief Review of Trigonometry
We begin with the xy-coordinate axes and the unit circle (the circle with radius 1 centered at the origin).
Starting at the point (1,0), if you move counter-clockwise along the
unit circle for a distance θ , then you’ll be at the point ( cos θ ,sin θ ) .
That last sentence defines both the cosine and the sine functions,
and the angle θ (in radians). θ is the actual length of an arc
(colored orange in the figure) along the unit circle starting at the point
(1,0), and the cosine and sine of θ are simply the x and y coordinates
of the terminating point of the arc.
Graphs of the sine and cosine functions
If you go in a clockwise direction instead of a
clockwise one, then θ is negative. The sine and
cosine functions are periodic with period 2π ,
simply because the circumference of the unit circle
is 2π , so going that distance along the circle takes
you right back to your starting point.
Trigonometry will have you going in circles!
All of the other trigonometric functions can be
defined in terms of the sine and cosine functions.
The tangent is defined to be the sine divided by the
cosine, the cotangent is the cosine divided by the
sine, the secant is the reciprocal of the cosine, and
the cosecant is the reciprocal of the sine:
tan θ =
sin θ
cos θ
cot θ =
cos θ
sin θ
sec θ =
1
cos θ
csc θ =
1
sin θ
Graphs of the tan, cot, sec, and csc
Can you see which graph is which ?
Since ( cos θ ,sin θ ) is a point on the unit circle, its distance from the origin is always 1. This yields the
famous Pythagorean identity:
cos 2 θ + sin 2 θ = 1
Dividing both sides of this identity by cos 2 θ yields the second of the three Pythagorean identities:
1 + tan 2 θ = sec 2 θ
Dividing both sides of the original identity by sin2θ yields the final Pythagorean identity:
cot 2 θ + 1 = csc 2 θ
Recall that an identity is true for all values of the variables, as contrasted with a typical equation, for
which only occasional special values of the variables satisfy it.
Geometry of the trig functions
All of the trigonometric functions have geometric
interpretations in terms of the unit circle,
as shown in the graph to the right:
In addition to the Pythagorean identities, there are many identities involving the trig functions. Some of
the ones most useful in calculus are:
sin( A + B) = sin A cos B + cos A sin B
cos( A + B ) = cos A cos B − sin A sin B
cos(2 A) = cos 2 A − sin 2 A
sin(2 A) = 2sin A ⋅ cos A
sin 2 A =
1 − cos ( 2 A )
2
cos 2 A =
1 + cos ( 2 A )
2
Two more identities basically state that the cosine is an even function, and the sine is an odd function:
cos(− x) = cos ( x )
&
sin(− x) = − sin ( x )
Sometimes it’s important to recognize the trigonometric functions in terms
of right triangles. From the triangle at the right, we have:
sin θ =
C
B
A
B
A
C
, cos θ = , tan θ = , cot θ = , sec θ = , csc θ =
B
C
C
A
B
A
Trigonometry and Calculus
You should know an important limit involving the sine function which you probably encountered during
your study of limits in Calculus I:
sin h
=1
lim
h →0
h
In particular, this was needed to calculate the derivative of the sine function. So in a sense, the entire
calculus of the trigonometric functions arises from this limit.
Calculus reveals many hidden connections between the trigonometric functions. Below are the
derivatives of the six basic trig functions (also from Calculus I), which reveal how the rates of change of
the trig functions are related to the other trig functions.
d
sin x = cos x
dx
d
cos x = − sin x
dx
d
tan x = sec 2 x
dx
d
cot x = − csc 2 x
dx
d
sec x = sec x tan x
dx
d
csc x = − csc x cot x
dx
Each of these derivative equations can be read “backwards” as a corresponding integral equation:
∫ cos x dx = sin x + C
∫ sin x dx = − cos x + C
∫ sec
∫ csc
2
x dx = tan x + C
∫ sec x tan x dx = sec x + C
2
x dx = − cot x + C
∫ csc x cot x dx = − csc x + C
Taking the derivative of both sides of an identity yields another identity. For example,
d
d
sin(2 x) = 2sin x ⋅ cos x
dx
dx
⇒
⇒
⎛⎛ d
d
⎞
⎞
2 cos(2 x) = 2 ⎜ ⎜ sin x ⎟ ⋅ cos x + sin x ⋅ cos x ⎟
dx
⎠
⎝ ⎝ dx
⎠
2 cos(2 x) = 2 ( cos x ⋅ cos x + sin x ⋅ ( − sin x ) ) = 2 ( cos 2 x − sin 2 x )
⇒
cos(2 x) = cos 2 x − sin 2 x