Trigonometric Functions on Any Angle

1
Trigonometric Functions on Any Angle

• Section 4.4

2
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.

3
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

4
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 p/2.
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5
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 2p.
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6
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.
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7
Reference Angle
There is a straight-forward process for finding
reference angles.
Step 3 Calculate the reference angle using the
quadrant-specific directions.
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8
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 ?.
Thus
where the reference angle is
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9
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 of the angle to p.
Thus
This distance is the reference angle.
Note Here put subtracted the angle from p since
the angle was smaller than p. This gave us the
positive reference angle. If we had subtracted p
from the angle, we would have needed to take the
absolute value of the answer.
continued on next slide
10
Reference Angle
Directions for quadrant III
This distance is the reference angle.
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 p to the
terminal side of the angle.
Thus
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11
Reference Angle
Directions for quadrant III
Note Here put subtracted p from the angle
since the angle was larger than p. This gave us
the positive reference angle. If we had
subtracted the angle from p, we would have needed
to take the absolute value of the answer.
continued on next slide
12
Reference Angle
Directions for quadrant IV
This distance is the reference angle.
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 2p.
Thus
continued on next slide
13
Reference Angle
Directions for quadrant IV
Note Here put subtracted the angle from 2p
since the angle was smaller than 2p. This gave
us the positive reference angle. If we had
subtracted 2p from the angle, we would have
needed to take the absolute value of the answer.
continued on next slide
14
Reference Angle Summary
Step 1 Find the angle coterminal to the given
angle that is between 0 and 2p.
Step 2 Determine the quadrant in which the
terminal side of the angle falls.
Step 3 Calculate the reference angle using the
quadrant-specific directions indicated to the
right.
15
In which quadrant is the angle ?
To find out what quadrant ? is in, we need to
determine which direction to go and how far.
Since the angle is negative, we need to go in the
clockwise direction. The distance we need to go
is one whole p and 1/6 of a p further.
This red part is approximately 1/6 of a p further.
This blue part is one whole p in the clockwise
direction
Now that we have drawn the angle, we can see that
the angle ? is in quadrant II.
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16
What is the reference angle, , for the angle
?
Using our summary for finding a reference angle,
we start by finding an angle coterminal to ? that
is between 0 and 2p. Thus we need to start by
adding 2p to our angle.
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17
What is the reference angle, , for the angle
?
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.
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18
What is the reference angle, , for the angle
?
Quadrant II
Finally we need to use the quadrant II directions
for finding the reference angle.
Thus the reference angle is
19
Evaluate each of the following for .
To solve a problem like this, we want to start by
finding the reference angle for ?.
Since our angle is bigger than 2p, we need to
subtract 2p to find the coterminal angle that is
between 0 and 2p.
continued on next slide
20
Evaluate each of the following for .
To find the reference angle for an angle in
quadrant II, we subtract the coterminal angle
from p.
This will give us a reference angle of
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21
Evaluate each of the following for .
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22
Evaluate each of the following for .
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.
continued on next slide
23
Evaluate each of the following for .
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.
continued on next slide
24
Evaluate each of the following for .
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.
25
For , find the values of the
trigonometric functions based on .
26
Evaluate the following expressions if
and
27
Evaluate the following expressions if
and
28
If and ? is in quadrant IV, then
find the following.