Trigonometric Functions on
Any Angle
Section 4.4
Objectives
• Determine the quadrant in which the terminal
side of an angle occurs.
• Find the reference angle of a given angle.
• Determine the sine, cosine, tangent,
cotangent, secant, and cosecant values of an
angle given one of the sine, cosine, tangent,
cotangent, secant, or cosecant value of the
angle.
Vocabulary
• quadrant
• reference angle
• sine of an angle
• cosine of an angle
• terminal side of an angle
• initial side of an angle
• tangent of an angle
• cotangent of an angle
• secant of an angle
• cosecant of an angle
Reference Angle
A reference angle is the
smallest distance between
the terminal side of an angle
and the x-axis.
All reference angles
will be between 0 and
π/2.
continued on next slide
Reference Angle
There is a straight-forward
process for finding
reference angles.
Step 1 – Find the
angle coterminal to
the given angle that is
between 0 and 2π.
continued on next slide
Reference Angle
There is a straight-forward
process for finding
reference angles.
Step 2 – Determine
the quadrant in which
the terminal side of
the angle falls.
continued on next slide
Reference Angle
There is a straight-forward
process for finding
reference angles.
Step 3 – Calculate the
reference angle using
the quadrant-specific
directions.
continued on next slide
Reference Angle
Directions for quadrant I
For quadrant I, the
shortest distance from
the terminal side of the
angle to the x-axis is the
same as the angle θ.
θ Thus
where the
reference
angle is
continued on next slide
Reference Angle
Directions for quadrant II
For quadrant II, the
shortest distance from
the terminal side of the
angle to the x-axis is
shown in blue. This is
the rest of the distance
from the terminal side θ
of the angle to π.
Thus This distance is the
reference angle.
Note: Here put subtracted the angle from π
since the angle was smaller than π. This gave us
the positive reference angle. If we had
subtracted π from the angle, we would have
needed to take the absolute value of the answer. continued on next slide
Reference Angle
Directions for quadrant III
This distance is the
reference angle.
θ
For quadrant III, the
shortest distance from
the terminal side of the
angle to the x-axis is
shown in blue. This is the
distance from the π to
the terminal side of the
angle. Thus
continued on next slide
Reference Angle
Directions for quadrant III
Note: Here put subtracted π from the angle
since the angle was larger than π. This gave us
the positive reference angle. If we had
subtracted the angle from π, we would have
needed to take the absolute value of the answer.
θ
continued on next slide
Reference Angle
Directions for quadrant IV
This distance is the
reference angle.
θ
For quadrant IV, the
shortest distance from
the terminal side of the
angle to the x-axis is
shown in blue. This is
the rest of the distance
from the terminal side
of the angle to 2π.
Thus
2
continued on next slide
Reference Angle
Directions for quadrant IV
Note: Here put subtracted
the angle from 2π since the
angle was smaller than 2π.
This gave us the positive
reference angle. If we had
subtracted 2π from the
angle, we would have needed θ
to take the absolute value
of the answer.
2
continued on next slide
Reference Angle Summary
Step 1 – Find the
angle coterminal to
the given angle that is
between 0 and 2π.
Quadrant II Quadrant I
Step 2 – Determine
the quadrant in which
the terminal side of
the angle falls.
Step 3 – Calculate the Quadrant III Quadrant IV
reference angle using
the quadrant-specific 2
directions indicated
to the right.
In which quadrant is the angle
7
6
?
To find out what quadrant
θ is in, we need to This red part is
determine which direction approximately 1/6 of a
to go and how far. Since π further.
the angle is negative, we
need to go in the clockwise
direction. The distance we
need to go is one whole π
and 1/6 of a π further.
This blue part is
Now that we have one whole π in the
drawn the angle, we clockwise direction
can see that the angle
θ is in quadrant II.
continued on next slide
What is the
reference
angle, , for
the angle
7
?
6
Using our summary for
finding a reference angle,
we start by finding an angle
coterminal to θ that is
between 0 and 2π. Thus we
need to start by adding 2π
to our angle.
7
a coterminal angle 2
6
7 12
a coterminal angle
6 6
5
a coterminal angle
6 continued on next slide
What is the
reference
angle, , for
the angle
7
?
6
5 5
a coterminal angle
6 6
The next step is to
determine what quadrant our
coterminal angle is in. We
really already did this in the
first question of the problem.
Coterminal angles always
terminate in the same
quadrant. Thus our
coterminal angle is in
quadrant II.
continued on next slide
What is the reference angle, ,
for the angle 7 ?
6
5
a coterminal angle
6
Quadrant II
Finally we need to use
the quadrant II
directions for finding the
reference angle.
5
6
6 5
6 6
6
Thus the reference angle is
6
Evaluate each of the following
11
for .
4
1. sin
To solve a problem like this, we
want to start by finding the
reference angle for θ.
Since our angle is bigger than 2π, we
need to subtract 2π to find the
coterminal angle that is between 0
and 2π.
11 11 8 3
2
4 4 4 4
continued on next slide
Evaluate each of the following
11
for .
4
1. sin
Our next step is to figure out
what quadrant
3 is in. You can 3
4 4
see from the picture that we are
in quadrant II.
To find the reference angle for an
angle in quadrant II, we subtract the
coterminal angle from π.
This will give us a reference angle
of 3 4 3
4 4 4 4 continued on next slide
Evaluate each of the following
11
for .
4
1. sin
We will now use the basic
trigonometric function values for 3
4 4
The only thing that we will need to
change might be the signs of the
basic values. Remember that the
sign of the cosine and tangent
functions will be negative in
quadrant II. The sign of the sine
will still be positive in quadrant II.
11 2
sin
4 2
continued on next slide
Evaluate each of the following
11
for .
4
2. cos
Once again, we will use our
reference angle to determine the
basic trigonometric function value.
The only difference between the
basic value and the value for our
angle may be the sign.
11 2
cos
4 2
continued on next slide
Evaluate each of the following
11
for .
4
3. tan
Once again, we will use our
reference angle to determine the
basic trigonometric function value.
The only difference between the
basic value and the value for our
angle may be the sign.
11 2
sin
2
11 4 1
tan
4 cos 11 2
4 2
continued on next slide
Evaluate each of the following
11
for .
4
4. sec
Once again, we will use our
reference angle to determine the
basic trigonometric function value.
The only difference between the
basic value and the value for our
angle may be the sign.
11 1 1 2
sec 2
4 cos 11 2 2
4 2
For 0 2 , find the values of
the trigonometric functions
10
based on csc .
9
1. sin
2. cos
3. tan
4. sec
5. cot
Evaluate the following
2
expressions if cos
7
and tan 0
1. sin
2. csc
3. tan
4. sec
5. cot
Evaluate the following
4
expressions if tan
3
and sin 0
1. sin
2. cos
3. cot
4. sec
5. csc
6
If cos and θ is in quadrant
8
IV, then find the following.
1. tan cot
2. csc tan
3. sin cos
2 2