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Objectives
Use the formula for the
cosine of the difference of
two angles.
Use sum and difference
formulas for cosines and sines.
Use sum and difference
formulas for tangents.
Sum and Difference Formulas
L
isten to the same note played on a piano and a
violin. The notes have a different quality or
“tone.” Tone depends on the way an instrument
vibrates. However, the less than 1% of the popula-
tion with amusia, or true tone deafness, cannot tell
the two sounds apart. Even simple,
familiar tunes such as Happy Birthday
and Jingle Bells are mystifying to
amusics.
When a note is played, it vibrates
at a specific fundamental frequency
and has a particular amplitude.
Amusics cannot tell the difference
between two sounds from tuning forks modeled by
and respectively.
However, they can recognize the difference
between the two equations. Notice that the second
equation contains the sine of the sum of two angles.
In this section, we will be developing identities
involving the sums or differences of two angles.
These formulas are called the sum and difference
formulas. We begin with the cosine of
the difference of two angles.
cos 1 a - b 2 ,
p = 3 sin 2t p = 2 sin 12 t + p 2 ,
S e c t i o n 5.
596 Chapter 5 Analytic Trigonometry
Critical Thinking Exercises
Make Sense? In Exercises 88–91, determine whether each statement makes sense or does not make sense, and explain your reasoning.
88. The word identity is used in different ways in additive identity, multiplicative identity, and trigonometric identity. 89. To prove a trigonometric identity, I select one side of the equation and transform it until it is the other side of the equation, or I manipulate both sides to a common trigono- metric expression. 90. In order to simplify I need to know how
to subtract rational expressions with unlike denominators.
91. The most efficient way that I can simplify
is to immediately rewrite the
expression in terms of cosines and sines.
In Exercises 92–95, verify each identity.
92.
96. Use one of the fundamental identities in the box on page 586 to create an original identity.
ln etan
(^2) x - sec 2 x ln ƒ^ sec x ƒ^ = -ln ƒ^ cos x ƒ = - 1
sin x - cos x + 1 sin x + cos x - 1
sin x + 1 cos x
sin^3 x - cos^3 x sin x - cos x
= 1 + sin x cos x
1 sec x + 121 sec x - 12 sin^2 x
cos x 1 - sin x
sin x cos x
Group Exercise
97. Group members are to write a helpful list of items for a pamphlet called “The Underground Guide to Verifying Identities.” The pamphlet will be used primarily by students who sit, stare, and freak out every time they are asked to verify an identity. List easy ways to remember the fundamen- tal identities. What helpful guidelines can you offer from the perspective of a student that you probably won’t find in math books? If you have your own strategies that work particularly well, include them in the pamphlet.
Preview Exercises Exercises 98–100 will help you prepare for the material covered in the next section.
98. Give exact values for cos 30°, sin 30°, cos 60°, sin 60°, cos 90°, and sin 90°. 99. Use the appropriate values from Exercise 98 to answer each of the following. a. Is or cos 90°, equal to b. Is or cos 90°, equal to 100. Use the appropriate values from Exercise 98 to answer each of the following. a. Is or sin 90°, equal to b. Is or sin 90°, equal to sin 30° cos 60° + cos 30° sin 60°?
sin 1 30° + 60° 2 ,
sin 1 30° + 60° 2 , sin 30° + sin 60°?
cos 30° cos 60° - sin 30° sin 60°?
cos 1 30° + 60° 2 ,
cos 1 30° + 60° 2 , cos 30° + cos 60°?
Section 5.2 Sum and Difference Formulas 597
The Cosine of the Difference of Two Angles
The cosine of the difference of two angles equals the cosine of the first angle
times the cosine of the second angle plus the sine of the first angle times the sine
of the second angle.
cos 1 a - b 2 = cos a cos b + sin a sin b
We use Figure 5.1 to prove the identity in the box. The graph in Figure 5.1(a)
shows a unit circle, The figure uses the definitions of the cosine and
sine functions as the and of points along the unit circle. For exam-
ple, point corresponds to angle By definition, the of is
and the is Similarly, point corresponds to angle By
definition, the of is and the is
Note that if we draw a line segment between points and a triangle is
formed. Angle is one of the angles of this triangle. What happens if we rotate
this triangle so that point falls on the at (1, 0)? The result is shown in
Figure 5.1(b). This rotation changes the coordinates of points and However, it
has no effect on the length of line segment
We can use the distance formula, to find an
expression for in Figure 5.1(a) and in Figure 5.1(b). By equating the two
expressions for we will obtain the identity for the cosine of the difference of
two angles, a - b.We first apply the distance formula in Figure 5.1(a).
PQ,
PQ
d = 41 x 2 - x 122 + 1 y 2 - y 122 ,
PQ.
P Q.
P x-axis
a - b
P Q,
x-coordinate Q cos a y-coordinate sin a.
y-coordinate sin b. Q a.
P b. x-coordinate P cos b
x- y-coordinates
x^2 + y^2 = 1.
a (^) b
a − b
x
y
x
y
a − b
P = (1, 0)
x^2 + y^2 = 1
Q = (cos α, sin α) P = (cos β, sin β)
Q = (cos (α − β), sin (α − β))
x^2 + y^2 = 1
(a)
(b) Figure 5.1 Using the unit circle and to develop a formula for cos 1 a - b 2
PQ
Apply the distance formula, to find the distance between and
Square each expression using
Regroup terms to apply a Pythagorean identity. Because each expression in parentheses equals 1.
= 22 - 2 cos a cos b - 2 sin a sin b Simplify.
= 21 + 1 - 2 cos a cos b - 2 sin a sin b sin^2 x + cos^2 x = 1,
= 41 sin^2 a + cos^2 a 2 + 1 sin^2 b + cos^2 b 2 - 2 cos a cos b - 2 sin a sin b
1 A - B 2^2 = A^2 - 2AB + B^2.
= 3 cos^2 a - 2 cos a cos b + cos^2 b + sin^2 a - 2 sin a sin b + sin^2 b
1 cos b , sin b 2 1 cos a , sin a 2.
d = 41 x 2 - x 1 2^2 + 1 y 2 - y 1 2^2 ,
PQ = 41 cos a - cos b 22 + 1 sin a - sin b 22
Next, we apply the distance formula in Figure 5.1(b) to obtain a second expression for
PQ.We let 1 x 1 , y 12 = 1 1, 0 2 and 1 x 2 , y 22 = 1 cos 1 a - b 2 , sin 1 a - b 22.
Apply the distance formula to find the distance between (1, 0) and
Square each expression.
Use a Pythagorean identity.
= 42 - 2 cos 1 a - b 2 Simplify.
=cos^2 (a-b)-2 cos (a-b)+1+sin^2 (a-b)
=1-2 cos (a-b)+
Using a Pythagorean identity, sin^2 (a − b) + cos^2 (a − b) = 1.
= 4 cos^21 a - b 2 - 2 cos 1 a - b 2 + 1 + sin^21 a - b 2
1 cos 1 a - b 2 , sin 1 a - b 22.
PQ = 43 cos 1 a - b 2 - 142 + 3 sin 1 a - b 2 - 042
The Cosine of the Difference of Two Angles
Section 5.2 Sum and Difference Formulas 599
Verifying an Identity
Verify the identity:
Solution We work with the left side.
Use the formula for
Divide each term in the numerator by
This step can be done mentally. We wanted you to see the substitutions that follow. Use quotient identities.
Simplify.
We worked with the left side and arrived at the right side. Thus, the identity is
verified.
Check Point 3 Verify the identity:
The difference formula for cosines is used to establish other identities. For
example, in our work with right triangles, we noted that cofunctions of complements
are equal. Thus, because and are complements,
We can use the formula for to prove this cofunction identity for all
angles.
Sum and Difference Formulas for Cosines and Sines
Our formula for can be used to verify an identity for a sum involving
cosines, as well as identities for a sum and a difference for sines.
cos 1 a - b 2
= sin u
= 0 #^ cos u + 1 #^ sin u
Apply cos (a − b) with a = and b = u. cos (a − b) = cos a cos b + sin a sin b
cos p
p
p
a -u b^ =cos cos^ u+sin^ sin^ u
π 2
cos 1 a - b 2
cosa
p
u
p
cos 1 a - b 2
cos a cos b
= 1 + tan a tan b.
= cot a + tan b
= cot a #^1 + 1 #^ tan b
cos a
sin a
cos^ b
cos b
sin a
sin a
sin^ b
cos b
sin a cos b.
cos a cos b
sin a cos b
sin a sin b
sin a cos b
cos 1 a - b 2.
cos 1 a - b 2
sin a cos b
cos a cos b + sin a sin b
sin a cos b
cos 1 a - b 2
sin a cos b
= cot a + tan b.
EXAMPLE 3
Technology
Graphic Connections
The graphs of
and
are shown in the same viewing rectangle. The graphs are the same. The displayed math on the right with the voice balloon on top shows the equivalence algebraically.
y = cos (^) ( π 2 − x ) and
1
−p p − 1
y = sin x
y = sin x
y = cosa
p 2
Use sum and difference formulas
for cosines and sines.
Sum and Difference Formulas for Cosines and Sines
4. sin 1 a - b 2 = sin a cos b - cos a sin b
sin 1 a + b 2 = sin a cos b + cos a sin b
cos 1 a - b 2 = cos a cos b + sin a sin b
cos 1 a + b 2 = cos a cos b - sin a sin b
600 Chapter 5 Analytic Trigonometry
Up to now, we have concentrated on the second formula in the box on the
previous page,. The first identity gives a
formula for the cosine of the sum of two angles. It is proved as follows:
Express addition as subtraction of an additive inverse. Use the difference formula for cosines.
Cosine is even: Sine is odd:
Simplify.
Thus, the cosine of the sum of two angles equals the cosine of the first angle times
the cosine of the second angle minus the sine of the first angle times the sine of the
second angle.
The third identity in the box gives a formula for the sine of the
sum of two angles. It is proved as follows:
Use a cofunction identity:
Regroup.
Use the difference formula for cosines.
Use cofunction identities.
Thus, the sine of the sum of two angles equals the sine of the first angle times the
cosine of the second angle plus the cosine of the first angle times the sine of the
second angle.
The final identity in the box, gives a
formula for the sine of the difference of two angles. It is proved
by writing as and then using the formula for the sine of
a sum.
Using the Sine of a Sum to Find an Exact Value
Find the exact value of using the fact that
Solution We apply the formula for the sine of a sum.
Substitute exact values.
Simplify.
Check Point (^) 4 Find the exact value of using the fact that
5 p
p
p
sin
5 p
22
22
= sin sin 1 a + b 2 = sin a cos b + cos a sin b
p
cos
p
+ cos
p
sin
p
sin
7 p
= sina
p
p
b
7 p
p
p
sin.
7 p
EXAMPLE 4
sin 1 a - b 2 sin 3 a + 1 - b 24
sin 1 a - b 2 ,
sin 1 a - b 2 = sin a cos b - cos a sin b,
= sin a cos b + cos a sin b.
= cosa
p
p
= cosB a
p
sin u = cos a
p 2
sin 1 a + b 2 = cosc
p
sin 1 a + b 2 ,
= cos a cos b - sin a sin b.
sin 1 - b 2 = - sin b.
= cos a cos b + sin a 1 - sin b 2 cos 1 - b 2 = cos b.
= cos a cos 1 - b 2 + sin a sin 1 - b 2
cos 1 a + b 2 = cos 3 a - 1 - b 24
cos 1 a - b 2 = cos a cos b + sin a sin b
602 Chapter 5 Analytic Trigonometry
d. We use the formula for the sine of a sum.
Check Point 5 Suppose that^ for a quadrant II angle^ and
for a quadrant I angle Find the exact value of each of the following:
a. b. c. d.
Verifying Observations on a Graphing Utility
Figure 5.4 shows the graph of in a by
viewing rectangle.
a. Describe the graph using another equation.
b. Verify that the two equations are equivalent.
Solution
a. The graph appears to be the cosine curve It cycles through
maximum, intercept, minimum, intercept, and back to maximum. Thus,
also describes the graph.
b. We must show that
We apply the formula for the sine of a difference on the left side.
and
Simplify.
This verifies our observation that and describe
the same graph.
Check Point 6 Figure 5.5^ shows the graph of^ in a
by viewing rectangle.
a. Describe the graph using another equation.
b. Verify that the two equations are equivalent.
Sum and Difference Formulas for Tangents
By writing as the quotient of and we can
develop a formula for the tangent of a sum. Writing subtraction as addition of an
inverse leads to a formula for the tangent of a difference.
tan 1 a + b 2 sin 1 a + b 2 cos 1 a + b 2 ,
c0, 2p,
p
y = cosax + d
3 p
b
y = sinax - y = cos x
3 p
b
= cos x
sin
3 p 2
cos = - 1
3 p 2
= sin x #^0 - cos x 1 - 12 = 0
sin a cos b - cos a sin b
sin 1 a - b 2 =
sinax -
3 p
b = sin x cos
3 p
3 p
sinax -
3 p
b = cos x.
y = cos x
y = cos x.
c0, 2p, 3 - 2, 2, 1 4
p
y = sinax - d
3 p
b
EXAMPLE 6
cos a cos b cos 1 a + b 2 sin 1 a + b 2.
b.
sin a = 45 a sin b = 12
4
+ a -
b #^
sin 1 a + b 2 = sin a cos b + cos a sin b
Figure 5.4 The graph of in a by 3 - 2, 2, 1 4 viewing rectangle
c0, 2p,
p y = sinax - 2 d
3 p 2 b
Figure 5.
Use sum and difference
formulas for tangents.
These values are given. These are the values we found.
sin a= , sin b=
cos a=– , cos b=
Section 5.2 Sum and Difference Formulas 603
Discovery
Derive the sum and difference formulas for tangents by working Exercises 55 and 56 in Exercise Set 5.2.
Sum and Difference Formulas for Tangents
The tangent of the sum of two angles equals the tangent of the first angle plus the
tangent of the second angle divided by 1 minus their product.
The tangent of the difference of two angles equals the tangent of the first angle
minus the tangent of the second angle divided by 1 plus their product.
tan 1 a - b 2 =
tan a - tan b
1 + tan a tan b
tan 1 a + b 2 =
tan a + tan b
1 - tan a tan b
Verifying an Identity
Verify the identity:
Solution We work with the left side.
Check Point 7 Verify the identity:^ tan^1 x^ +^ p^2 =^ tan^ x.
tan x - 1
tan x + 1
tan
p 4
tan x - 1
1 + tan x #^1
tan 1 a - b 2 =
tan a - tan b 1 + tan a tan b
tanax -
p
b =
tan x - tan
p
1 + tan x tan
p
tanax -
p
b =
tan x - 1
tan x + 1
EXAMPLE 7
Exercise Set 5.
Practice Exercises
Use the formula for the cosine of the difference of two angles to solve Exercises 1–12.
In Exercises 1–4, find the exact value of each expression.
**1. 2.
- 4.**
In Exercises 5–8, each expression is the right side of the formula for with particular values for and
a. Identify and in each expression. b. Write the expression as the cosine of an angle. c. Find the exact value of the expression. 5.
6. cos 50° cos 5° + sin 50° sin 5°
cos 50° cos 20° + sin 50° sin 20°
a b
cos 1 a - b 2 a b.
cosa
2 p 3
p 6
cosa b
3 p 4
p 6
b
cos 1 45° - 30° 2 cos 1 120° - 45° 2
In Exercises 9–12, verify each identity.
12. cosax -
5 p 4
b = -
1 cos x + sin x 2
cosax -
p 4
b =
1 cos x + sin x 2
cos 1 a - b 2 sin a sin b
= cot a cot b + 1
cos 1 a - b 2 cos a sin b
= tan a + cot b
cos
5 p 18
cos
p 9
5 p 18
sin
p 9
cos
5 p 12
cos
p 12
5 p 12
sin
p 12
Section 5.2 Sum and Difference Formulas 605
In Exercises 65–68, the graph with the given equation is shown in a
by viewing rectangle.
a. Describe the graph using another equation. b. Verify that the two equations are equivalent.
65.
Practice Plus
In Exercises 69–74, rewrite each expression as a simplified expression containing one term.
69.
70.
sin 1 a + b 2 - sin 1 a - b 2 cos 1 a + b 2 + cos 1 a - b 2
sin 1 a - b 2 cos b + cos 1 a - b 2 sin b
cos 1 a + b 2 cos b + sin 1 a + b 2 sin b
y = cosax -
p 2
b - cosax +
p 2
b
y = sinax +
p 2
b + sina
p 2
y = cos 1 x - 2 p 2
y = sin 1 p - x 2
c0, 2p, 3 - 2, 2, 1 4
p 2
d
(Do not use four different identities to solve this exercise.)
(Do not use four different identities to solve this exercise.)
In Exercises 75–78, half of an identity and the graph of this half are given. Use the graph to make a conjecture as to what the right side of the identity should be. Then prove your conjecture. 75.
[− 2 p, 2p, q] by [−2, 2, 1]
cos
5 x 2
cos 2x + sin
5 x 2
sin 2x =?
[− 2 p, 2p, q] by [−2, 2, 1]
sin
5 x 2
cos 2x - cos
5 x 2
sin 2x =?
[− 2 p, 2p, q] by [−2, 2, 1]
sin 5x cos 2x - cos 5x sin 2x =?
[− 2 p, 2p, q] by [−2, 2, 1]
cos 2x cos 5x + sin 2x sin 5x =?
sina
p 3
p 3
p 3
p 3
cosa
p 6
p 6
p 6
p 6
cos 1 a - b 2 + cos 1 a + b 2
- sin 1 a - b 2 + sin 1 a + b 2
606 Chapter 5 Analytic Trigonometry
Application Exercises
79. A ball attached to a spring is raised 2 feet and released with an initial vertical velocity of 3 feet per second. The distance of the ball from its rest position after seconds is given by Show that
where lies in quadrant I and Use the identity to find the amplitude and the period of the ball’s motion.
80. A tuning fork is held a certain distance from your ears and struck. Your eardrums’ vibrations after seconds are given by When a second tuning fork is struck, the formula describes the effects of the sound on the eardrums’ vibrations. The total vibrations are given by
a. Simplify to a single term containing the sine.
b. If the amplitude of is zero, no sound is heard. Based on your equation in part (a), does this occur with the two tuning forks in this exercise? Explain your answer.
Writing in Mathematics
In Exercises 81–86, use words to describe the formula for each of the following:
81. the cosine of the difference of two angles. 82. the cosine of the sum of two angles. 83. the sine of the sum of two angles. 84. the sine of the difference of two angles. 85. the tangent of the difference of two angles. 86. the tangent of the sum of two angles. 87. The distance formula and the definitions for cosine and sine are used to prove the formula for the cosine of the difference of two angles. This formula logically leads the way to the other sum and difference identities. Using this development of ideas and formulas, describe a characteristic of mathematical logic.
Technology Exercises
In Exercises 88–93, graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates that the equation is not an identity. In these exercises, find a value of for which both sides are defined but not equal.
88.
89. tan 1 p - x 2 = -tan x
cosa
3 p 2
x
p
p
p = 3 sin 2t + 2 sin 12 t + p 2.
p = 2 sin 12 t + p 2
p = 3 sin 2t.
t
u tan u = 32.
2 cos t + 3 sin t = 2 13 cos 1 t - u 2 ,
d = 2 cos t + 3 sin t.
t
Critical Thinking Exercises Make Sense? In Exercises 94–97, determine whether each statement makes sense or does not make sense, and explain your reasoning.
94. I’ve noticed that for sine, cosine, and tangent, the trig function for the sum of two angles is not equal to that trig function of the first angle plus that trig function of the second angle. 95. After using an identity to determine the exact value of sin 105°, I verified the result with a calculator. 96. Using sum and difference formulas, I can find exact values for sine, cosine, and tangent at any angle. 97. After the difference formula for cosines is verified, I noticed that the other sum and difference formulas are verified relatively quickly. 98. Verify the identity:
In Exercises 99–102, find the exact value of each expression. Do not use a calculator.
99.
In Exercises 103–105, write each trigonometric expression as an algebraic expression (that is, without any trigonometric functions). Assume that and are positive and in the domain of the given inverse trigonometric function.
103.
104.
105. tan 1 sin-^1 x + cos-^1 y 2
sin 1 tan-^1 x - sin-^1 y 2
cos 1 sin-^1 x - cos-^1 y 2
x y
cosBcos-^1 ¢ -
≤ - sin-^1 a -
b R
cosatan-^1
b
sinBsin-^1
b R
sinacos-^1
b
sin 1 x - y 2 cos x cos y
sin 1 y - z 2 cos y cos z
sin 1 z - x 2 cos z cos x
sin 1.2x cos 0.8x + cos 1.2x sin 0.8x = sin 2x
cos 1.2x cos 0.8x - sin 1.2x sin 0.8x = cos 2x
cosax +
p 2
b = cos x + cos
p 2
sinax +
p 2
b = sin x + sin
p 2
