Table of Contents |
The unit circle is a circle of radius 1 that is centered at the origin. Consider the graph below.
Using the Pythagorean theorem, 
, or more simply, 
. This is true for any point on the unit circle. The equation of the unit circle is .
Next, consider the figure below, which shows the unit circle and other information:
a point on the circle.
to 
the central angle of the circular arc.
Every point 
corresponds to the length s of some circular arc. This means that x and y can be defined as functions of s, meaning s is the input, and x and y are outputs.
and represents the x-coordinate of the point. That is, 
and represents the y-coordinate of the point. That is, 
If 
then there is no length, which means no movement from the point 
Thus, every real number s is paired with one point on the graph of the unit circle. This means that the domain of the sine and cosine functions is the set of all real numbers, or 
Now, how does the length s relate to the angle 
Since 
and 
we have 
Thus, we can also express the sine and cosine function in terms of the angle, as well as its length.
To summarize, any point on the unit circle can be expressed as 
where 
is the central angle of the arc; or 
where s is the length of the arc. As you progress through this course, you will notice that using the angle as input becomes necessary to make more general definitions.
or 
respectively.
and 
but just like the logarithmic functions, we usually only use grouping symbols when there is more than a single variable as the input.
is the input, you will write 
as opposed to 
Some textbooks use the latter, and this is mathematically correct, but it is always best to emphasize the input by placing grouping symbols around it.
EXAMPLE
Consider the point P
which is a point on the unit circle since it satisfies the equation 
be the angle of inclination from the positive x-axis to the line connecting the origin to P, as shown in the figure.
and 
Let’s say that angle terminates at some ray that intersects the unit circle at some point Then, any angle coterminal to will also terminate at the same ray, which means the values of sine and cosine are the same as they were at the angle 
This results in the following property.
and 
Since a sine or cosine function can be applied to any angle, the domain of both functions is the set of all real numbers.
Considering all possible points on the unit circle, we see that 
and 
Since 
and 
for any point on the unit circle, this means that 
and 
and 
Each function has domain 
and range 
When substituting certain values of into the sine and cosine functions, we can refer to the unit circle to find these values. These special angles are:
| Degrees |
|
|
|
|
|
|---|---|---|---|---|---|
| Radians | 0 |
|
|
|
|
Let’s say we want the value of the sine function when the angle is 
You would write 
when the angle is in radian form, which is equivalent to 
when the angle is in degree form.
To start, let’s consider 
radians, which is also 
Since 
and 
when 
it follows that 
and 
Along the same line of thinking, let’s now consider 
or in degrees, 
Since 
and 
when it follows that 
and 
Now, let’s find the values at 
and 
which are and 
respectively.
and 
Now, consider the angle 
or in degrees, 
Since it is easier to visualize, we’ll use degree measures to illustrate.
Consider this figure.
We know the following:
since the sum of the angles in a triangle is 
By using the Pythagorean theorem, we can find the value of x.
|
This is the Pythagorean theorem. |
|
Substitute the quantities. Since 1 is the hypotenuse,  then other sides are each x.
|
|
Combine like terms. |
|
Divide both sides by 2. |
|
Apply the square root principle. |
|
Since the point is in the first quadrant, use the positive solution. |
|
Rewrite as a quotient of square roots and simplify the numerator. |
|
Rationalize the denominator. |
This means that 
Since 
it follows that 
Thus, the terminal point on the arc of the circle is 
which means 
and 
In radian form, 
and 
To summarize, we have the following table, which organizes the values of the sine and cosine functions for each special angle.
 in Degrees
|
|
|
|
|
|
|---|---|---|---|---|---|
|
in Radians
|
0 |
|
|
|
|
|
1 |
|
|
|
0 |
|
0 |
|
|
|
1 |
Consider this picture below, which shows the signs of the coordinates in each quadrant:
Since 
and 
we can make the following observations:
in quadrants I and IV and 
in quadrants II and III. 
in quadrants I and II and 
in quadrants III and IV.
and are positive in quadrant I and are both negative in quadrant III.
To evaluate trigonometric functions for angles that terminate outside the first quadrant, we need to use reference angles.
For the purposes of this exploration, let’s assume that angle terminates in quadrant I, and corresponds to the point 
where x and y are positive. Our goal is to determine a relationship between and the angles that correspond to the points 
in quadrant II, 
in quadrant III, and 
in quadrant IV.
First, let’s examine a quadrant II angle.
Let 
be an angle that terminates in quadrant II, and corresponds to the point 
as shown in the figure.
The cosines of the angles are opposites of each other since the x-coordinates are opposites. The sines of the angles are equal since the y-coordinates are equal.
Now notice the angle 
which has the same measure as 
and is the smallest positive angle between and the x-axis. If the coordinates are known in the first quadrant, then using which is the reference angle, tells us the values of the coordinates in the second quadrant, since they have the same magnitude, but possibly different sign.
Seeing this now, here is a picture that shows what’s happening for angles that terminate in each quadrant. The formulas below each graph are used to calculate the reference angle in both degree form and in radian form.
|
|
|
|
In each case, 
is the reference angle. Note that the measure of is the same as the angle between the positive x-axis and the ray in the first quadrant each time.
EXAMPLE
Find the exact value of
is 
in degree form. The angle is sketched below.
terminates in quadrant II, the reference angle is 
, or
it follows that 
since x-coordinates are negative in quadrant II.
EXAMPLE
Find the exact value of
is 
in degree form. The angle is sketched below.
terminates in quadrant III, the reference angle is 
or 
and y-coordinates are negative in quadrant III, 
Here is one more example, this time using an angle that terminates in quadrant IV.
EXAMPLE
Evaluate
is equivalent to 
which means that it terminates in quadrant IV. A sketch of the angle is shown below.
terminates in quadrant IV, the reference angle is 
it follows that 
since x-coordinates are positive in quadrant IV.
Using the unit circle, we can define four more trigonometric functions, all based on the angle and point on the unit circle:
is written 
It is the ratio of the y-coordinate to the x-coordinate. That is, 
where 
is written 
It is the ratio of the x-coordinate to the y-coordinate. That is, 
where 
Notice that this is the reciprocal of the tangent function. is written 
It is the reciprocal of the x-coordinate. That is, 
where Notice that this is the reciprocal of the cosine function. is written 
It is the reciprocal of the y-coordinate. That is, 
where Notice that this is the reciprocal of the sine function. on the unit circle, and angle that inclines from the positive x-axis to the line with endpoints 
and we have the following definitions of the remaining four trigonometric functions:
where 
where 
where
where
With these definitions, let’s revert back to the unit circle to understand the domain of these functions. We’ll explore the range of these functions when we examine their graphs later in this course.
To examine the domains of these trigonometric functions, the information is organized in the chart.
| Trigonometric Function and Its Ratio From the Unit Circle | Undefined at These Points on the Unit Circle |
Undefined for These Values of
|
Domain of Function |
|---|---|---|---|
|
where
|
 and 
|
 plus all coterminal angles
|
All real numbers except for odd multiples of
|
|
where
|
and 
|
 plus all coterminal angles
|
All real numbers except all integer multiples of 
|
|
where
|
and
|
plus all coterminal angles
|
All real numbers except for odd multiples of
|
|
where
|
and
|
plus all coterminal angles
|
All real numbers except all integer multiples of
|
Taking all the formulas for these trigonometric functions and replacing x with and y with we have the following identities.
where 
where 
where
where
where 
and 
is definedThere are six trigonometric functions total, but three of them are reciprocals of the other three. In practice, it is easier to find the values of sine, cosine, and tangent functions, then take their reciprocals to get the values of the cosecant, secant, and cotangent functions (as long as values are not undefined).
With all these definitions, we can find the values of all six trigonometric functions of an angle
EXAMPLE
Consider the point P
which is a point on the unit circle since it satisfies the equation
where the terminal side of intersects the unit circle at the point 
Using the values of and at special angles, we can also find values of the other trigonometric functions.
EXAMPLE
Find the exact values of
and 
|
Use the identity 
|
|
 
|
|
Simplify; multiply the numerator and denominator by 2. |
|
Rationalize the denominator. |
|
Use the identity 
|
|
|
|
Simplify. |
|
Rationalize the denominator and simplify. |
Using the relationships between the trigonometric functions and the coordinates on the unit circle, we can tell what the sign of the function value is simply by knowing the quadrant in which the angle terminates.
For example, consider the fact that 
If x and y have the same sign, then 
If x and y have different signs, then 
If we repeat this process for all six trigonometric functions and their defined ratios, we could summarize the results this way, listing where the functions return positive values.
|
Quadrant II (-, +) |
Quadrant I (+, +) |
|---|---|
|
All functions are positive. |
|
Quadrant III (-, -) |
Quadrant IV (+, -) |
 
|
|
If you want a mnemonic to remember this by, just remember All Students Take Calculus!
EXAMPLE
Find the exact value of
This means that the angle terminates in quadrant II, and has reference angle 
This means that the value of 
is negative.
|
|
Use the identity
|
|
|
|
|
|
Simplify; multiply the numerator and denominator by 2. |
|
|
Rationalize the denominator. |
terminates in quadrant III, so its reference angle is 
which means that secant is the reciprocal of cosine.
it follows that 
since the cosine of an angle in quadrant III is always negative.
is the reciprocal of 
which is 
Since it is preferred to not have radicals in the denominator, we rationalize and simplify:
Consider the figure shown below, which shows that angle corresponds to the point , while the angle 
corresponds to the point 
Let’s compare and 
Since the value of the sine function is the y-coordinate of the point, we have and 
In general, we can say that 
Is this true for all the trigonometric functions?
Let’s compare and 
Since the value of the cosine function is the x-coordinate of the point, we have and 
In general, we can say that 
Extending this to the other trigonometric functions, we can establish identities known as the even/odd identities.
Given a function 
recall that 
is even if 
and odd if 
Therefore, the sine, cosecant, tangent, and cotangent functions are odd, while the cosine and secant functions are even.
These identities are useful to evaluate trigonometric functions of negative angles when the function value at the positive angle is known.
EXAMPLE
first.
to rationalize the denominator to get 
The even/odd identities are very useful when graphing trigonometric functions, which is covered in a future lesson.
SOURCE: THIS WORK IS ADAPTED FROM PRECALCULUS BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/BOOKS/PRECALCULUS/PAGES/1-INTRODUCTION-TO-FUNCTIONS
where 
where 
where
If k is an integer, then 
and 
where
where 
and 
is defined
where 
where 
where 
where 
