Double-Angle and Half-Angle Identities

1
Double-Angle and Half-Angle Identities

• Section 5.3

2
Objectives

• Apply the half-angle and/or double angle formula
  to simplify an expression or evaluate an angle.
• Apply a power reducing formula to simplify an
  expression.

3
Double-Angle Identities
4
Half-Angle Identities
5
Power-Reducing Identities
6
Use a half-angle identity to find the exact
value of
We will use the half-angle formula for sine
We need to find out what a is in order to use
this formula.
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7
Use a half-angle identity to find the exact
value of
We now replace a with in the formula to
get
continued on next slide
8
Use a half-angle identity to find the exact
value of
Now all that we have left to do is determine if
the answer should be positive or negative.
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9
Use a half-angle identity to find the exact
value of
We determine which to use based on what quadrant
the original angle is in. In our case, we need
to know what quadrant p/12 is in. This angle
fall in quadrant I. Since the sine values in
quadrant I are positive, we keep the positive
answer.
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10
If find the values of the following
trigonometric functions.
For this we will need the double angle formula
for cosine
In order to use this formula, we will need the
cos(t) and sin(t). We can use either the
Pythagorean identity or right triangles to find
sin(t).
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11
If find the values of the following
trigonometric functions.
Triangle for angle t
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12
If find the values of the following
trigonometric functions.
Triangle for angle t
Since angle t is in quadrant III, the sine value
is negative.
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13
If find the values of the following
trigonometric functions.
Now we fill in the values for sine and cosine.
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14
If find the values of the following
trigonometric functions.
For this we will need the double angle formula
for cosine
We know the values of both sin(t) and cos(t)
since we found them for the first part of the
problem.
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15
If find the values of the following
trigonometric functions.
Now we fill in the values for sine and cosine.
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16
At this point, we can ask the question What
quadrant is the angle 2t in?
This question can be answered by looking at the
signs of the sin(2t) and cos(2t).
is positive
and
is positive
The only quadrant where both the sine value and
cosine value of an angle are positive is quadrant
I.
17
If find the values of the following
trigonometric functions.
We will use the half-angle formula for sine
Since we know the value of cos(t), we can just
plug that into the formula.
continued on next slide
18
If find the values of the following
trigonometric functions.
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19
If find the values of the following
trigonometric functions.
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20
If find the values of the following
trigonometric functions.
Now all that we have left to do is determine if
the answer should be positive or negative. In
order to do this, we need to know which quadrant
the angle t/2 falls in.
To do this we will need to use the information we
have about the angle t.
continued on next slide
21
If find the values of the following
trigonometric functions.
The information that we have about the angle t is
What we need is information about t/2. If we
divide each piece of the inequality, we will get
t/2 in the middle of the inequality and bounds
for the angle on the left and right sides.
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22
If find the values of the following
trigonometric functions.
Thus we see that the angle t/2 is in quadrant II.
continued on next slide
23
If find the values of the following
trigonometric functions.
Since the angle t/2 is in quadrant II, the sine
value must be positive.
continued on next slide
24
If find the values of the following
trigonometric functions.
We will use the half-angle formula for sine
Since we know the value of cos(t), we can just
plug that into the formula.
continued on next slide
25
If find the values of the following
trigonometric functions.
continued on next slide
26
If find the values of the following
trigonometric functions.
continued on next slide
27
If find the values of the following
trigonometric functions.
Now all that we have left to do is determine if
the answer should be positive or negative. In
order to do this, we need to know which quadrant
the angle t/2 falls in.
To do this we will need to use the information we
have about the angle t.
continued on next slide
28
If find the values of the following
trigonometric functions.
The information that we have about the angle t is
What we need is information about t/2. If we
divide each piece of the inequality, we will get
t/2 in the middle of the inequality and bounds
for the angle on the left and right sides.
continued on next slide
29
If find the values of the following
trigonometric functions.
Thus we see that the angle t/2 is in quadrant II.
continued on next slide
30
If find the values of the following
trigonometric functions.
Since the angle t/2 is in quadrant II, the cosine
value must be negative.
31
Use the power-reducing formula to simplify the
expression
We need to use the power-reducing identity for
the cosine and sine functions to do this problem.
In our problem the angle a in the formula will be
7x in our problem. We also need to rewrite our
problem.
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32
Use the power-reducing formula to simplify the
expression
Now we just apply the identity to get
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33
Use the power-reducing formula to simplify the
expression
This is much simpler than the original expression
and the power (exponent) is clearly reduced.
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34
Use the power-reducing formula to simplify the
expression
Is there another way to simplify this without
using a power-reducing formula?
The answer to this question is yes. The original
expression is the difference of two squares and
can be factoring into
Now you should notice that the expression in the
second set of square brackets is the Pythagorean
identity and thus is equal to 1.
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35
Use the power-reducing formula to simplify the
expression
Is there another way to simplify this without
using a power-reducing formula?
Now you should notice that what is left is the
right side of the double angle identity for
cosine where the angle a is 7x.
This will allow us to rewrite the expression as