Download 486 18–2 Sum or Difference of Two Angles

486
◆
Chapter 18
18–2
Trigonometric Identities and Equations
Sum or Difference of Two Angles
Sine of the Sum of Two Angles
We wish now to derive a formula for the sine of the sum of two angles, say, and . For example,
is it true that
sin 20 sin 30 sin 50?
Try it on your calculator—you will see that it is not true.
We start by drawing two positive acute angles, and (Fig. 18–2), small enough so that
their sum ( ) is also acute. From any point P on the terminal side of we draw perpendicular AP to the x axis, and draw perpendicular BP to line OB. Since the angle between two
lines equals the angle between the perpendiculars to those two lines, we note that angle APB is
equal to .
y
P
+
D
B
A
O
C
x
FIGURE 18–2
Then
AP AD PD BC PD
sin ( ) OP
OP
OP
BC PD
OP OP
But in triangle OBC,
BC OB sin and in triangle PBD,
PD PB cos Substituting, we obtain
OB sin PB cos sin ( ) OP
OP
But in triangle OPB,
OB
OP
= cos and
PB
OP
sin Section 18–2
◆
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Sum or Difference of Two Angles
Thus:
sin ( ) sin cos cos sin The sine of the sum of two angles is not the sum of the sines
of each angle.
Common
Error
sin ( ) sin sin Cosine of the Sum of Two Angles
Again using Fig. 8–2, we can derive an expression for cos( ). From Eq. 147,
OA OC AC OC BD
cos ( ) OP
OP
OP
OC BD
OP OP
Now, in triangle OBC,
OC OB cos and in triangle PDB,
BD PB sin Substituting, we obtain
OB cos PB sin cos ( ) OP
OP
As before,
OB
OP
cos and
PB
OP
sin Therefore:
cos ( ) cos cos sin sin Difference of Two Angles
We can obtain a formula for the sine of the difference of two angles merely by substituting for in the equation previously derived for sin( ).
sin[ ()] sin cos() cos sin()
But for in the first quadrant, () is in the fourth, so
cos() cos and
sin() sin Therefore,
sin ( ) sin cos cos (sin )
which is rewritten as follows:
sin ( ) sin cos cos sin We have proven this true for
acute angles whose sum is also
acute. These identities are, in
fact, true for any size angles,
positive or negative, although we
will not take the space to prove
this.
488
Chapter 18
◆
Trigonometric Identities and Equations
We see that the result is identical to the formula for sin( ) except for a change in sign. This
enables us to write the two identities in a single expression using the sign.
When the double signs are used
(such as the symbol), it is
understood that the upper signs
and the lower signs correspond
with the elements in the
equation.
Sine of Sum
or Difference
of Two Angles
sin( ) sin cos cos sin 167
Similarly, finding cos( ), we have
cos[ ()] cos cos() sin sin()
Thus:
cos ( ) cos cos sin sin or:
Cosine of Sum
or Difference
of Two Angles
◆◆◆
cos( ) cos cos sin sin 168
Example 9: Expand the expression sin (x 3y).
Solution: By Eq. 167,
sin (x 3y) sin x cos 3y cos x sin 3y
◆◆◆
◆◆◆
Example 10: Simplify
cos 5x cos 3x sin 5x sin 3x
Solution: We see that this has a similar form to Eq. 168, with 5x and 3x, so
cos 5x cos 3x sin 5x sin 3x cos (5x 3x)
cos 8x
◆◆◆
◆◆◆
Example 11: Prove that
cos(180 ) cos Solution: Expanding the left side by means of Eq. 168, we get
cos(180 ) cos 180 cos sin 180 sin But cos 180 1 and sin 180 0, so
cos(180 ) (1) cos (0) sin cos ◆◆◆
Example 12: Prove that
tan x tan y
sin (x y)
tan x tan y
sin (x y)
◆◆◆
Solution: The right side contains functions of the sum of two angles, but the left side contains
functions of single angles. We thus start by expanding the right side by using Eq. 167.
sin (x y)
sin x cos y cos x sin y
sin (x y)
sin x cos y cos x sin y
Our expression now contains sin x, sin y, cos x, and cos y, but we want an expression containing
only tan x and tan y.
Section 18–2
◆
489
Sum or Difference of Two Angles
To have tan x instead of sin x, we can divide numerator and denominator by cos x. Similarly,
to obtain tan y instead of sin y, we can divide by cos y. We thus divide numerator and denominator
by cos x cos y.
sin x cos y
cos x sin y
sin x cos y cos x sin y cos x cos y cos x cos y
sin x cos y cos x sin y
sin x cos y
cos x sin y
cos x cos y cos x cos y
Then, by Eq. 162,
tan x tan y
tan x tan y
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Tangent of the Sum or Difference of Two Angles
Since, by Eq. 162,
sin tan cos we simply divide Eq. 167 by Eq. 168.
sin( )
sin cos cos sin tan ( ) cos( ) cos cos sin sin Dividing numerator and denominator by cos cos yields
sin cos sin cos tan ( ) sin sin 1 cos cos Applying Eq. 162 again, we get
tan tan 1 tan tan A similar derivation (which we will not do) will show that tan( ) is identical to the expression just derived, except, as we might expect, for a reversal of signs. We combine the two
expressions using double signs as follows:
Tangent of Sum
or Difference of
Two Angles
◆◆◆
tan cos tan ( ) 1 tan tan 169
Example 13: Simplify
tan 3x tan 2x
tan 2x tan 3x 1
Solution: This can be put into the form of Eq. 169 if we factor (1) from the denominator.
tan 3x tan 2x
tan 2x tan 3x 1
tan 3x tan 2x
(tan 2x tan 3x 1)
tan 3x tan 2x
1 tan 3x tan 2x
tan (3x 2x)
tan 5x
◆◆◆
490
Chapter 18
◆◆◆
◆
Trigonometric Identities and Equations
Example 14: Prove that
1 tan x
tan (45 x) 1 tan x
Solution: Expanding the left side by Eq. 169, we get
tan 45 tan x
tan (45 x) 1 tan 45 tan x
But tan 45 1, so
1 tan x
tan(45 x) 1 tan x
◆◆◆
◆◆◆
Example 15: Prove that
cot y cot x
tan (x y)
cot x cot y 1
Solution: Using Eq. 152c on the left side yields
1
1
tan y tan x
1
tan x
1
tan y
1
Multiply numerator and denominator by tan x tan y.
tan x tan y
tan(x y)
1 tan x tan y
Then, by Eq. 169:
tan (x y) tan (x y)
◆◆◆
Adding a Sine Wave and a Cosine Wave of the Same Frequency
In Sec. 17–3, we used vectors to show that the sum of a sine wave and a cosine wave of the same
frequency could be written as a single sine wave, at the original frequency, but with some phase
angle. The resulting equation is useful in electrical applications. Here we use the formula for
the sum or difference of two angles to derive that equation.
Let A sin t be a sine wave of amplitude A, and B cos t be a cosine wave of amplitude B,
each of frequency /2. If we draw a right triangle (Fig. 18–3) with sides A and B and hypotenuse R, then
A R cos R
and
B R sin B = R sin A = R cos FIGURE 18–3
The sum of the sine wave and the cosine wave is then
A sin t B cos t R sin t cos R cos t sin R (sin t cos cos t sin )
the expression on the right will be familiar. It is the relationship of the sine of the sum of two
quantities, t and . Continuing, we have
A sin t B cos t R (sin t cos cos t sin )
R sin(t )
Section 18–2
◆
491
Sum or Difference of Two Angles
by Eq. 167. Thus we have the following equation:
Addition of a
Sine Wave and
Cosine Wave
◆◆◆
A sin t B cos t R sin (t )
where
B
R 兹 A2 B2 and arctan A
This is sometimes referred to as
magnitude and phase form.
214
Example 16: Express the following as a single sine function:
y 3.46 sin t 2.28 cos t
Solution: The magnitude of the resultant is
R 兹(3.46)2 (2.28)2 4.14
and the phase angle is
2.28
arctan arctan 0.659 33.4
3.46
So
y 3.46 sin t 2.28 cos t
4.14 sin(t 33.4)
Exercise 2
◆
Sum or Difference of Two Angles
Expand by means of the addition and subtraction formulas, and simplify.
1. sin( 30)
2. cos(45 x)
3. sin(x 60)
4. tan( )
5. cos p x q
2
6. tan(2x y)
7. sin( 2)
8. cos[( ) ]
9. tan(2 3)
Simplify.
10. cos 2x cos 9x sin 2x sin 9x
11. cos( ) sin( )
12. sin 3 cos 2 cos 3 sin 2
13. sin p x q cos p x q
3
6
Prove each identity.
14. cos x sin(x 90)
15. sin(30 x) cos(60 x) cos x
16. 2 sin cos sin( ) sin( )
17. sin( 60) cos( 30) 兹 3 cos 18. cos(2 x) cos x
◆◆◆
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Chapter 18
◆
Trigonometric Identities and Equations
19. sin cos(30 ) sin(60 )
20. cos(x y) cos(x y) 2 cos x cos y
21. cos x sin p x q p sin x q
6
6
22. tan(360 ) tan 23. cos(60 ) sin(330 ) 0
sin 4x cos 4x
24. sin 5x sec x csc x sin x
cos x
sin(x y) tan x tan y
25. sin(x y) tan x tan y
26. cos(x 60) cos(60 x) cos x
cos(x y)
27. tan y cot x
sin x cos y
28. cotp x q tanp x q 0
4
4
cos sin 29. tan( 45) cos sin cot cot 1
30. cot( )
cot cot 1 tan x
31. tan p x q
1 tan x
4
Express as a single sine function.
32. y 47.2 sin t 64.9 cos t
33. y 8470 sin t 7360 cos t
34. y 1.83 sin t 2.74 cos t
35. y 84.2 sin t 74.2 cos t
18–3
Functions of Double Angles
Sine of 2
An equation for the sine of 2 is easily derived by setting in Eq. 167.
sin( ) sin cos cos sin which is rewritten in the following form:
Sine of Twice an Angle
sin 2 2 sin cos 170
Cosine of 2
Similarly, setting in Eq. 168, we have
cos( ) cos cos sin sin which is also given as follows:
Cosine of Twice an Angle
cos 2 cos2 sin2 171a