Definition of the six trigonometric functions
We will begin by
considering an angle in standard position.
Definition An angle in standard position is an angle lying in the Cartesian plane whose vertex is at the origin and whose initial ray lies along the positive xaxis. 
The following diagram depicts an angle θ in standard position.
The rotation of an angle in standard position originates from the initial ray. Angles formed by counterclockwise rotation have positive measure, while angles
formed by clockwise rotation have negative measure as pictured above. Two angles in
standard position that have the same terminal ray are called coterminal. For example, the
angles in the above figure are coterminal.
Suppose we choose a point (x, y) ≠ (0, 0) lying on the terminal ray of an angle in
standard position. If we let r be the distance from the origin to the point (x, y)
then,
according to the distance formula. Using the variables x, y, and r, we define the
six trigonometric functions as follows,
The six trigonometric functions are read as follows:
Abbreviation 
Function 
cos θ 
cosine θ 
sin θ 
sine θ 
tan θ 
tangent θ 
sec θ 
secant θ 
csc θ 
cosecant θ 
cot θ 
cotangent θ 
There are two important points to notice as you study these definitions. First, the the secant, cosecant, and cotangent functions are the reciprocals of the cosine, sine, and tangent functions, respectively. Second, there is no value for which the cosine and sine functions are undefined. This is because r is the distant from the origin to the point (x,y) ≠ (0,0) on the terminal ray.
Now let's apply these definitions to a real situation. Consider the following angle in standard position:
Using the above diagram, we compute r as,
We can now find the values of the six trigonometric functions with x = −4, y = 3, and
r = 5 as,
Radian Measure
We have not specifically discussed the angle θ
yet, but it can be measured in
degrees or in radians. While you are probably comfortable with degree measure,
you may be less so with radian measure. To define a radian, consider the following
circle with radius r centered at the origin:
In the above picture, the angle θ has measure 1 radian. To be specific,
Definition The radian measure of an angle whose vertex lies at the center of a circle is the ratio of the
arc length to the radius of the circle. 
The radius and arc length in the above
picture are equal (both equal r), so θ = 1. Likewise, if the radius of a circle
is r and the central angle intercepts an arc length of 2r, then the central angle
measures 2r/r = 2 radians.
Converting between degrees and radians
The angle corresponding to one complete rotation has measure 360 ° or 2π radians.
To convert from degrees to radians or radians to degrees we simply use the
following conversion factors:
2π radians = 360°
or π radians = 180°.
For example, if we want to convert x° to y radians, we simply write,
On the other hand, if we want to convert y radians to x°, we simply write,
To look at a concrete example, suppose we want to convert 512° to radians. Since
we are going from degrees to radians, we set the conversion up so that degrees
cancel,
In general, we will work with radian measure. Unless the degree unit (°) is
explicitly written, all angles are assumed to be in radians.
Domain and Range of Trigonometric Functions
We will now discuss the domain and range of each trigonometric function. Remember that
elements in the domain are valid inputs, in this case angle measurements, and elements in the range are
the corresponding outputs. Outputs of the trigonometric functions are simply ratios of the
variables x, y, and r.
To begin, let’s look at the domain and range of the trigonometric functiony = cos θ and
y = sin θ . Both of these trigonometric functions have domain all real numbers and range
{y  − 1 ≤ y ≤ 1}. To determine the domain, we ask ourselves, "for what values
of θ are cos θ and sin θ defined?" To answer this, recall the definitions of cos θ and
sin θ,
The ratios x/r and y/r are well defined provided r ≠ 0. Recall that r measures the
distance from the point (x, y) ≠ (0, 0) lying on the terminal side of an angle in
standard position to the origin. By definition, r cannot be zero. Thus, all values
of θ are valid inputs.
To see that, {y − 1 ≤ y ≤ 1} is the range of y = cos θ and y = sin θ, we need to
inspect the ratios x/r and y/r . If we write r in terms of x and y we have,
Notice that Thus, cos θ = x/ r and
sin θ = y/r cannot be greater than 1 or less than −1, depending on whether x and
y are positive or negative (r = x when y = 0 and r = y when x = 0). So we
conclude that the range of cos θ and sin θ is,
{y  − 1 ≤ y ≤ 1} .
The other trigonometric functions, specifically tan θ , sec θ , csc θ ,
and cot θ, contain an additional statement, either
x ≠ 0 or y ≠ 0. We will use these restrictions to determine their domain and
range. We will begin with y = tan θ = y / x . Notice
that y / x is not defined when x = 0. Since x is the xcoordinate of a point lying
on the terminal ray of an angle in standard position, we need to remove angles
that correspond to points whose xcoordinate is zero. Points lying on the yaxis
have xcoordinate equal to zero, so we must remove angles whose terminal ray
lies along either the positive or negative yaxis. Two such angles are pictured
below:
These two angles measure 90° (along the positive yaxis) and 270° (along the
negative yaxis) corresponding to
π/2 and 3π/2 radians. We also need to
consider all coterminal angles by adding and subtracting multiples of 2π (e.g.,
π/2 + 2π = 5π/2 and
π/2 − 2π = −3π/2 ). Therefore, we conclude that y = tan θ has
domain,
The range of tan θ is easier to determine. Since y = tan θ = y / x , and y and x are
coordinates of a point, y can be any real number and x can be any real number
except 0. Thus, the ratio y / x can be any real number, and we conclude that the
range of y = tan θ is all real numbers.
The domain of y = sec θ = r / x is the same as the domain of y = tan θ= y / x since
x ≠ 0 in both cases,
Knowing that the range of y = cos θ = x / r is {y  − 1 ≤ y ≤ 1}, we can easily
find the range of
y = sec = r /
x , since cos θ and sec θ are reciprocals of one
another. Reciprocating changes the direction of
inequalities in the range and we have,
{y  y ≤ −1 or y ≥ 1}.
To find the domain of y = csc θ = r / y , we notice that this function is not defined
when y = 0. Since y is the ycoordinate of a point lying on the terminal ray of an
angle in standard position, we need to remove angles that correspond to points
whose ycoordinate is zero. Points lying on the xaxis have ycoordinate equal to
zero, so we must remove angles whose terminal ray lies along either the positive
or negative xaxis. Two such angles are pictured below:
These two angles measure 180° along the negative xaxis and 360° along the
positive xaxis corresponding to π and 2π radians, respectively. Of course we also need to
consider all coterminal angles by adding and subtracting multiples of 2π (e.g., π + 2π = 3π
and π − 2π = −π ). Therefore, we conclude that y = csc θ has
domain,
{ θ θ ≠ . . . ,−3π ,−2π ,−π , 0,π , 2π , 3π , . . .} .
Knowing that the range of y = sin θ= y / r is {y  − 1 ≤ y ≤ 1}, we can easily find
the range of
y = csc θ = r/ y ,
since sin θ and csc θ are reciprocals of one another. Thus, the range of y = csc θ is,
{y  y ≤ −1 or y ≥ 1} .
The domain of y = cot θ = x / y is the same as the domain of y = csc θ = r / y since
y ≠ 0 in both cases,
{θ  θ≠
. . . ,−3π ,−2π ,−π , 0,π , 2π , 3π , . . .}.
Using the same argument to find the range of y = tan θ, the range of y = cot θ is also all real numbers.
The domains and ranges of the six trigonometric functions are summarized in the following
table:
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In the next section we will find the trigonometric functional values of some special angles.
Special Angles
