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MATH 201 - Week 11
MATH 201 - Week 11
Ferenc Balogh
Concordia University
2008 Winter
Based on the textbook
J. Stuart, L. Redlin, S. Watson, Precalculus - Mathematics for Calculus, 5th Edition, Thomson
MATH 201 - Week 11
Overview
1 Addition and Subtraction Formulas - Section 7.2FormulasApplicationsA sin x + B cos x
2 Double-Angle, Half-Angle and Product-Sum Formulas - Section7.3
Squares of Trigonometric FunctionsHalf-Angle FormulasProduct-Sum Formulas
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Formulas
Addition Formulas
sin(s + t) = sin s cos t + cos s sin t
cos(s + t) = cos s cos t − sin s sin t
tan(s + t) =tan s + tan t
1− tan s tan t
Subtraction Formulas
sin(s − t) = sin s cos t − cos s sin t
cos(s − t) = cos s cos t + sin s sin t
tan(s − t) =tan s − tan t
1 + tan s tan t
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Formulas
Addition Formulas
sin(s + t) = sin s cos t + cos s sin t
cos(s + t) = cos s cos t − sin s sin t
tan(s + t) =tan s + tan t
1− tan s tan t
Subtraction Formulas
sin(s − t) = sin s cos t − cos s sin t
cos(s − t) = cos s cos t + sin s sin t
tan(s − t) =tan s − tan t
1 + tan s tan t
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Formulas
Example. Prove the addition formula for the tangent using theaddition formulas for the sine and cosine functions.
Solution.
tan(s + t) =sin(s + t)
cos(s + t)
=sin s cos t + cos s sin t
cos s cos t − sin s sin t
=cos s cos t
cos s cos t
tan s + tan t
1− tan s tan t
=tan s + tan t
1− tan s tan t.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Formulas
Example. Prove the addition formula for the tangent using theaddition formulas for the sine and cosine functions.Solution.
tan(s + t) =sin(s + t)
cos(s + t)
=sin s cos t + cos s sin t
cos s cos t − sin s sin t
=cos s cos t
cos s cos t
tan s + tan t
1− tan s tan t
=tan s + tan t
1− tan s tan t.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Formulas
Example. Prove the addition formula for the tangent using theaddition formulas for the sine and cosine functions.Solution.
tan(s + t) =sin(s + t)
cos(s + t)
=sin s cos t + cos s sin t
cos s cos t − sin s sin t
=cos s cos t
cos s cos t
tan s + tan t
1− tan s tan t
=tan s + tan t
1− tan s tan t.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Formulas
Example. Prove the addition formula for the tangent using theaddition formulas for the sine and cosine functions.Solution.
tan(s + t) =sin(s + t)
cos(s + t)
=sin s cos t + cos s sin t
cos s cos t − sin s sin t
=cos s cos t
cos s cos t
tan s + tan t
1− tan s tan t
=tan s + tan t
1− tan s tan t.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Formulas
Example. Prove the addition formula for the tangent using theaddition formulas for the sine and cosine functions.Solution.
tan(s + t) =sin(s + t)
cos(s + t)
=sin s cos t + cos s sin t
cos s cos t − sin s sin t
=cos s cos t
cos s cos t
tan s + tan t
1− tan s tan t
=tan s + tan t
1− tan s tan t.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Find the exact value of sin 5π12 and tan π
12 .
Solution. Since 5π12 = 3π
12 + 2π12 = π
4 + π6 we have
sin5π
12= sin
(π4
+π
6
)= sin
(π4
)cos(π
6
)+ cos
(π4
)sin(π
6
)=
1√2
√3
2+
1√2
1
2=
√3 + 1
2√
2.
Similarly, using π12 = π
3 −π4 we get
tan( π
12
)=
tan π3 − tan π
4
1 + tan π3 tan π
4
=
√3− 1
1 +√
3 · 1=
√3− 1
1 +√
3.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Find the exact value of sin 5π12 and tan π
12 .
Solution. Since 5π12 = 3π
12 + 2π12 = π
4 + π6 we have
sin5π
12= sin
(π4
+π
6
)= sin
(π4
)cos(π
6
)+ cos
(π4
)sin(π
6
)=
1√2
√3
2+
1√2
1
2=
√3 + 1
2√
2.
Similarly, using π12 = π
3 −π4 we get
tan( π
12
)=
tan π3 − tan π
4
1 + tan π3 tan π
4
=
√3− 1
1 +√
3 · 1=
√3− 1
1 +√
3.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Find the exact value of sin 5π12 and tan π
12 .
Solution. Since 5π12 = 3π
12 + 2π12 = π
4 + π6 we have
sin5π
12= sin
(π4
+π
6
)= sin
(π4
)cos(π
6
)+ cos
(π4
)sin(π
6
)=
1√2
√3
2+
1√2
1
2=
√3 + 1
2√
2.
Similarly, using π12 = π
3 −π4 we get
tan( π
12
)=
tan π3 − tan π
4
1 + tan π3 tan π
4
=
√3− 1
1 +√
3 · 1=
√3− 1
1 +√
3.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Find the exact value of
sin 20◦ cos 70◦ + cos 20◦ sin 70◦.
Solution.
sin 20◦ cos 70◦ + cos 20◦ sin 70◦ = sin(20◦ + 70◦)
= sin 90◦ = 1.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Find the exact value of
sin 20◦ cos 70◦ + cos 20◦ sin 70◦.
Solution.
sin 20◦ cos 70◦ + cos 20◦ sin 70◦ = sin(20◦ + 70◦)
= sin 90◦ = 1.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Prove the cofunction identity
sin(π
2− x)
= cos x .
Solution.
LHS = sin(π
2− x)
= sinπ
2cos x − cos
π
2sin x
= 1 · cos x − 0 · sin x
= cos x = RHS .
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Prove the cofunction identity
sin(π
2− x)
= cos x .
Solution.
LHS = sin(π
2− x)
= sinπ
2cos x − cos
π
2sin x
= 1 · cos x − 0 · sin x
= cos x = RHS .
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Prove the cofunction identity
sin(π
2− x)
= cos x .
Solution.
LHS = sin(π
2− x)
= sinπ
2cos x − cos
π
2sin x
= 1 · cos x − 0 · sin x
= cos x = RHS .
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Prove the cofunction identity
sin(π
2− x)
= cos x .
Solution.
LHS = sin(π
2− x)
= sinπ
2cos x − cos
π
2sin x
= 1 · cos x − 0 · sin x
= cos x = RHS .
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Prove the identity
sin(x + y)− sin(x − y) = 2 cos x sin y .
Solution.
LHS = sin(x + y)− sin(x − y)
= (sin x cos y + cos x sin y)− (sin x cos y − cos x sin y)
= sin x cos y + cos x sin y − sin x cos y + cos x sin y
= 2 cos x sin y = RHS .
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Prove the identity
sin(x + y)− sin(x − y) = 2 cos x sin y .
Solution.
LHS = sin(x + y)− sin(x − y)
= (sin x cos y + cos x sin y)− (sin x cos y − cos x sin y)
= sin x cos y + cos x sin y − sin x cos y + cos x sin y
= 2 cos x sin y = RHS .
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Prove the identity
sin(x + y)− sin(x − y) = 2 cos x sin y .
Solution.
LHS = sin(x + y)− sin(x − y)
= (sin x cos y + cos x sin y)− (sin x cos y − cos x sin y)
= sin x cos y + cos x sin y − sin x cos y + cos x sin y
= 2 cos x sin y = RHS .
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
Applications
Example. Prove the identity
sin(x + y)− sin(x − y) = 2 cos x sin y .
Solution.
LHS = sin(x + y)− sin(x − y)
= (sin x cos y + cos x sin y)− (sin x cos y − cos x sin y)
= sin x cos y + cos x sin y − sin x cos y + cos x sin y
= 2 cos x sin y = RHS .
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
Example. Write the expression
√3
2sin x +
1
2cos x
in terms of sine only.
Solution. Notice that
cosπ
6=
√3
2, sin
π
6=
1
2.
Therefore√
3
2sin x +
1
2cos x = cos
π
6sin x + sin
π
6cos x
= sin(x +
π
6
).
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
Example. Write the expression
√3
2sin x +
1
2cos x
in terms of sine only.
Solution. Notice that
cosπ
6=
√3
2, sin
π
6=
1
2.
Therefore√
3
2sin x +
1
2cos x = cos
π
6sin x + sin
π
6cos x
= sin(x +
π
6
).
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
Example. Write the expression
√3
2sin x +
1
2cos x
in terms of sine only.
Solution. Notice that
cosπ
6=
√3
2, sin
π
6=
1
2.
Therefore√
3
2sin x +
1
2cos x = cos
π
6sin x + sin
π
6cos x
= sin(x +
π
6
).
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
In general, an expression of the form
A sin x + B cos x ,
where A and B are real numbers and not both of them are zero, can beexpressed in terms of sine only.
To apply the trick above, we have to find an angle φ such that
cosφ =A√
A2 + B2sinφ =
B√A2 + B2
.
Then
A sin x + B cos x =√
A2 + B2
(A√
A2 + B2sin x +
B√A2 + B2
cos x
)=
√A2 + B2 (cosφ sin x + sinφ cos x)
=√
A2 + B2 sin(x + φ).
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
In general, an expression of the form
A sin x + B cos x ,
where A and B are real numbers and not both of them are zero, can beexpressed in terms of sine only.To apply the trick above, we have to find an angle φ such that
cosφ =A√
A2 + B2sinφ =
B√A2 + B2
.
Then
A sin x + B cos x =√
A2 + B2
(A√
A2 + B2sin x +
B√A2 + B2
cos x
)=
√A2 + B2 (cosφ sin x + sinφ cos x)
=√
A2 + B2 sin(x + φ).
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
In general, an expression of the form
A sin x + B cos x ,
where A and B are real numbers and not both of them are zero, can beexpressed in terms of sine only.To apply the trick above, we have to find an angle φ such that
cosφ =A√
A2 + B2sinφ =
B√A2 + B2
.
Then
A sin x + B cos x =√
A2 + B2
(A√
A2 + B2sin x +
B√A2 + B2
cos x
)=
√A2 + B2 (cosφ sin x + sinφ cos x)
=√
A2 + B2 sin(x + φ).
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
A sin x + B cos x
A sin x + B cos x = k sin(x + φ),
wherek =
√A2 + B2
and
cosφ =A√
A2 + B2, sinφ =
B√A2 + B2
.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
Example. Write the expression
12 sin x + 5 cos x
in terms of sine only. What are the period, amplitude and phase shift?
Solution. In this example, A = 12, B = 5 and hence
k =√
A2 + B2 =√
122 + 52 =√
144 + 25 =√
169 = 13.
12 sin x + 5 cos x = 13
(12
13sin x +
5
13cos x
)= 13 sin(x + φ),
where
cosφ =12
13, sinφ =
5
13.
So the amplitude is 13, the period is 2π and the phase shift is given by
φ = arccos12
13≈ 22.62◦.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
Example. Write the expression
12 sin x + 5 cos x
in terms of sine only. What are the period, amplitude and phase shift?
Solution. In this example, A = 12, B = 5 and hence
k =√
A2 + B2 =√
122 + 52 =√
144 + 25 =√
169 = 13.
12 sin x + 5 cos x = 13
(12
13sin x +
5
13cos x
)= 13 sin(x + φ),
where
cosφ =12
13, sinφ =
5
13.
So the amplitude is 13, the period is 2π and the phase shift is given by
φ = arccos12
13≈ 22.62◦.
MATH 201 - Week 11
Addition and Subtraction Formulas - Section 7.2
A sin x + B cos x
Example. Write the expression
12 sin x + 5 cos x
in terms of sine only. What are the period, amplitude and phase shift?
Solution. In this example, A = 12, B = 5 and hence
k =√
A2 + B2 =√
122 + 52 =√
144 + 25 =√
169 = 13.
12 sin x + 5 cos x = 13
(12
13sin x +
5
13cos x
)= 13 sin(x + φ),
where
cosφ =12
13, sinφ =
5
13.
So the amplitude is 13, the period is 2π and the phase shift is given by
φ = arccos12
13≈ 22.62◦.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Double-Angle Formulas
sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x
= 1− 2 sin2 x
= 2 cos2 x − 1
tan 2x =2 tan x
1− tan2 x
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Proofs.We use the addition formulas:
sin 2x = sin(x + x)
= sin x cos x + cos x sin x
= 2 sin x cos x .
cos 2x = cos(x + x)
= cos x cos x − sin x sin x
= cos2 x − sin2 x .
tan 2x = tan(x + x)
=tan x + tan x
1− tan x tan x
=2 tan x
1− tan2 x.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Proofs.We use the addition formulas:
sin 2x = sin(x + x)
= sin x cos x + cos x sin x
= 2 sin x cos x .
cos 2x = cos(x + x)
= cos x cos x − sin x sin x
= cos2 x − sin2 x .
tan 2x = tan(x + x)
=tan x + tan x
1− tan x tan x
=2 tan x
1− tan2 x.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Proofs.We use the addition formulas:
sin 2x = sin(x + x)
= sin x cos x + cos x sin x
= 2 sin x cos x .
cos 2x = cos(x + x)
= cos x cos x − sin x sin x
= cos2 x − sin2 x .
tan 2x = tan(x + x)
=tan x + tan x
1− tan x tan x
=2 tan x
1− tan2 x.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Proofs.We use the addition formulas:
sin 2x = sin(x + x)
= sin x cos x + cos x sin x
= 2 sin x cos x .
cos 2x = cos(x + x)
= cos x cos x − sin x sin x
= cos2 x − sin2 x .
tan 2x = tan(x + x)
=tan x + tan x
1− tan x tan x
=2 tan x
1− tan2 x.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Example. Find sin 2x , cos 2x and tan 2x if
sin x = −3
5
and x belongs to the third quadrant.
Solution. We have to find the value of cos x first:
cos x = ±√
1− sin2 x = ±√
1− 9
25= ±
√16
25= ±4
5.
The sign must be negative because x belongs to the third quadrant.Therefore tan x = 3
4 and
sin 2x = 2 sin x cos x = 2 ·(−3
5
)·(−4
5
)=
24
25
cos 2x = cos2 x − sin2 x =16
25− 9
25=
7
25
tan 2x =2 tan x
1− tan2 x=
2 · 34
1− 916
=64716
=6
4· 16
7=
24
7.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Example. Find sin 2x , cos 2x and tan 2x if
sin x = −3
5
and x belongs to the third quadrant.
Solution. We have to find the value of cos x first:
cos x = ±√
1− sin2 x = ±√
1− 9
25= ±
√16
25= ±4
5.
The sign must be negative because x belongs to the third quadrant.Therefore tan x = 3
4 and
sin 2x = 2 sin x cos x = 2 ·(−3
5
)·(−4
5
)=
24
25
cos 2x = cos2 x − sin2 x =16
25− 9
25=
7
25
tan 2x =2 tan x
1− tan2 x=
2 · 34
1− 916
=64716
=6
4· 16
7=
24
7.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Example. Find sin 2x , cos 2x and tan 2x if
sin x = −3
5
and x belongs to the third quadrant.
Solution. We have to find the value of cos x first:
cos x = ±√
1− sin2 x = ±√
1− 9
25= ±
√16
25= ±4
5.
The sign must be negative because x belongs to the third quadrant.Therefore tan x = 3
4 and
sin 2x = 2 sin x cos x = 2 ·(−3
5
)·(−4
5
)=
24
25
cos 2x = cos2 x − sin2 x =16
25− 9
25=
7
25
tan 2x =2 tan x
1− tan2 x=
2 · 34
1− 916
=64716
=6
4· 16
7=
24
7.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Example. Find sin 2x , cos 2x and tan 2x if
sin x = −3
5
and x belongs to the third quadrant.
Solution. We have to find the value of cos x first:
cos x = ±√
1− sin2 x = ±√
1− 9
25= ±
√16
25= ±4
5.
The sign must be negative because x belongs to the third quadrant.Therefore tan x = 3
4 and
sin 2x = 2 sin x cos x = 2 ·(−3
5
)·(−4
5
)=
24
25
cos 2x = cos2 x − sin2 x =16
25− 9
25=
7
25
tan 2x =2 tan x
1− tan2 x=
2 · 34
1− 916
=64716
=6
4· 16
7=
24
7.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
Squares of Trigonometric Functions
sin2 x =1− cos 2x
2
cos2 x =1 + cos 2x
2
tan2 x =1− cos 2x
1 + cos 2x.
These are used to lower the powers of the trigonometric functionsin simplifying certain expressions.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
Example. Lower the powers of the trigonometric functions in thefollowing expression:
sin4 x cos2 x .
Solution.
sin4 x cos2 x =(sin2 x
)2cos2 x
=
(1− cos 2x
2
)21 + cos 2x
2
=
(1− 2 cos 2x + cos2 2x
4
)1 + cos 2x
2
=1
8
(1− 2 cos 2x +
1 + cos 4x
2
)(1 + cos 2x)
=1
16(3− 4 cos 2x + cos 4x) (1 + cos 2x)
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
Example. Lower the powers of the trigonometric functions in thefollowing expression:
sin4 x cos2 x .
Solution.
sin4 x cos2 x =(sin2 x
)2cos2 x
=
(1− cos 2x
2
)21 + cos 2x
2
=
(1− 2 cos 2x + cos2 2x
4
)1 + cos 2x
2
=1
8
(1− 2 cos 2x +
1 + cos 4x
2
)(1 + cos 2x)
=1
16(3− 4 cos 2x + cos 4x) (1 + cos 2x)
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
Example. Lower the powers of the trigonometric functions in thefollowing expression:
sin4 x cos2 x .
Solution.
sin4 x cos2 x =(sin2 x
)2cos2 x
=
(1− cos 2x
2
)21 + cos 2x
2
=
(1− 2 cos 2x + cos2 2x
4
)1 + cos 2x
2
=1
8
(1− 2 cos 2x +
1 + cos 4x
2
)(1 + cos 2x)
=1
16(3− 4 cos 2x + cos 4x) (1 + cos 2x)
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
Example. Lower the powers of the trigonometric functions in thefollowing expression:
sin4 x cos2 x .
Solution.
sin4 x cos2 x =(sin2 x
)2cos2 x
=
(1− cos 2x
2
)21 + cos 2x
2
=
(1− 2 cos 2x + cos2 2x
4
)1 + cos 2x
2
=1
8
(1− 2 cos 2x +
1 + cos 4x
2
)(1 + cos 2x)
=1
16(3− 4 cos 2x + cos 4x) (1 + cos 2x)
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
Example. Lower the powers of the trigonometric functions in thefollowing expression:
sin4 x cos2 x .
Solution.
sin4 x cos2 x =(sin2 x
)2cos2 x
=
(1− cos 2x
2
)21 + cos 2x
2
=
(1− 2 cos 2x + cos2 2x
4
)1 + cos 2x
2
=1
8
(1− 2 cos 2x +
1 + cos 4x
2
)(1 + cos 2x)
=1
16(3− 4 cos 2x + cos 4x) (1 + cos 2x)
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
Example. Lower the powers of the trigonometric functions in thefollowing expression:
sin4 x cos2 x .
Solution.
sin4 x cos2 x =(sin2 x
)2cos2 x
=
(1− cos 2x
2
)21 + cos 2x
2
=
(1− 2 cos 2x + cos2 2x
4
)1 + cos 2x
2
=1
8
(1− 2 cos 2x +
1 + cos 4x
2
)(1 + cos 2x)
=1
16(3− 4 cos 2x + cos 4x) (1 + cos 2x)
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
=1
16
(3− 4 cos 2x + cos 4x + 3 cos 2x − 4 cos2 2x + cos 4x cos 2x
)
=1
16(3− cos 2x + cos 4x − 2(1 + cos 4x) + cos 4x cos 2x)
=1
16(1− cos 2x − cos 4x + cos 4x cos 2x) .
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
=1
16
(3− 4 cos 2x + cos 4x + 3 cos 2x − 4 cos2 2x + cos 4x cos 2x
)=
1
16(3− cos 2x + cos 4x − 2(1 + cos 4x) + cos 4x cos 2x)
=1
16(1− cos 2x − cos 4x + cos 4x cos 2x) .
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Squares of Trigonometric Functions
=1
16
(3− 4 cos 2x + cos 4x + 3 cos 2x − 4 cos2 2x + cos 4x cos 2x
)=
1
16(3− cos 2x + cos 4x − 2(1 + cos 4x) + cos 4x cos 2x)
=1
16(1− cos 2x − cos 4x + cos 4x cos 2x) .
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Half-Angle Formulas
Squares of Trigonometric Functions
sinu
2= ±
√1− cos u
2
cosu
2= ±
√1 + cos u
2
tanu
2=
1− cos u
sin u
=sin u
1 + cos u.
The signs are determined by the quadrant in which the terminalpoint of u
2 lies.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Half-Angle Formulas
Example. Find the exact value of sin 9π8 .
Solution.
sin9π
8= ±
√1− cos 9π
4
2.
The angle 9π8 is in the third quadrant so its sine is negative and
cos9π
4= cos
(π4
+ 2π)
= cosπ
4=
1√2.
Hence
sin9π
8= −
√1− cos 9π
4
2
= −
√1− 1√
2
2= −
√√2− 1
2√
2
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Half-Angle Formulas
Example. Find the exact value of sin 9π8 .
Solution.
sin9π
8= ±
√1− cos 9π
4
2.
The angle 9π8 is in the third quadrant so its sine is negative and
cos9π
4= cos
(π4
+ 2π)
= cosπ
4=
1√2.
Hence
sin9π
8= −
√1− cos 9π
4
2
= −
√1− 1√
2
2= −
√√2− 1
2√
2
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Half-Angle Formulas
Example. Find the exact value of sin 9π8 .
Solution.
sin9π
8= ±
√1− cos 9π
4
2.
The angle 9π8 is in the third quadrant so its sine is negative and
cos9π
4= cos
(π4
+ 2π)
= cosπ
4=
1√2.
Hence
sin9π
8= −
√1− cos 9π
4
2
= −
√1− 1√
2
2= −
√√2− 1
2√
2
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Product-to-Sum Formulas
sin u cos v =1
2
(sin(u + v) + sin(u − v)
)cos u sin v =
1
2
(sin(u + v)− sin(u − v)
)cos u cos v =
1
2
(cos(u + v) + cos(u − v)
)sin u sin v =
1
2
(cos(u + v)− cos(u − v)
).
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Prove the product-to-sum formula
cos u sin v =1
2
(sin(u + v)− sin(u − v)
).
Solution. Using the addition formulas we get
RHS =1
2
(sin(u + v)− sin(u − v)
)=
1
2
((sin u cos v + cos u sin v)− (sin u cos v − cos u sin v)
)=
1
2
(sin u cos v + cos u sin v − sin u cos v + cos u sin v
)=
1
2
(2 cos u sin v
)= cos u sin v = LHS .
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Prove the product-to-sum formula
cos u sin v =1
2
(sin(u + v)− sin(u − v)
).
Solution. Using the addition formulas we get
RHS =1
2
(sin(u + v)− sin(u − v)
)
=1
2
((sin u cos v + cos u sin v)− (sin u cos v − cos u sin v)
)=
1
2
(sin u cos v + cos u sin v − sin u cos v + cos u sin v
)=
1
2
(2 cos u sin v
)= cos u sin v = LHS .
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Prove the product-to-sum formula
cos u sin v =1
2
(sin(u + v)− sin(u − v)
).
Solution. Using the addition formulas we get
RHS =1
2
(sin(u + v)− sin(u − v)
)=
1
2
((sin u cos v + cos u sin v)− (sin u cos v − cos u sin v)
)
=1
2
(sin u cos v + cos u sin v − sin u cos v + cos u sin v
)=
1
2
(2 cos u sin v
)= cos u sin v = LHS .
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Prove the product-to-sum formula
cos u sin v =1
2
(sin(u + v)− sin(u − v)
).
Solution. Using the addition formulas we get
RHS =1
2
(sin(u + v)− sin(u − v)
)=
1
2
((sin u cos v + cos u sin v)− (sin u cos v − cos u sin v)
)=
1
2
(sin u cos v + cos u sin v − sin u cos v + cos u sin v
)
=1
2
(2 cos u sin v
)= cos u sin v = LHS .
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Prove the product-to-sum formula
cos u sin v =1
2
(sin(u + v)− sin(u − v)
).
Solution. Using the addition formulas we get
RHS =1
2
(sin(u + v)− sin(u − v)
)=
1
2
((sin u cos v + cos u sin v)− (sin u cos v − cos u sin v)
)=
1
2
(sin u cos v + cos u sin v − sin u cos v + cos u sin v
)=
1
2
(2 cos u sin v
)
= cos u sin v = LHS .
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Prove the product-to-sum formula
cos u sin v =1
2
(sin(u + v)− sin(u − v)
).
Solution. Using the addition formulas we get
RHS =1
2
(sin(u + v)− sin(u − v)
)=
1
2
((sin u cos v + cos u sin v)− (sin u cos v − cos u sin v)
)=
1
2
(sin u cos v + cos u sin v − sin u cos v + cos u sin v
)=
1
2
(2 cos u sin v
)= cos u sin v = LHS .
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Write sin 2x cos 3x as a sum of trigonometric functions.
Solution. Using the first product-to-sum formula with u = 2x ,v = 3x we have
sin 2x cos 3x =1
2
(sin(2x + 3x) + sin(2x − 3x)
)=
1
2
(sin 5x + sin(−x)
)=
1
2
(sin 5x − sin x
).
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Write sin 2x cos 3x as a sum of trigonometric functions.
Solution. Using the first product-to-sum formula with u = 2x ,v = 3x we have
sin 2x cos 3x =1
2
(sin(2x + 3x) + sin(2x − 3x)
)
=1
2
(sin 5x + sin(−x)
)=
1
2
(sin 5x − sin x
).
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Write sin 2x cos 3x as a sum of trigonometric functions.
Solution. Using the first product-to-sum formula with u = 2x ,v = 3x we have
sin 2x cos 3x =1
2
(sin(2x + 3x) + sin(2x − 3x)
)=
1
2
(sin 5x + sin(−x)
)
=1
2
(sin 5x − sin x
).
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Write sin 2x cos 3x as a sum of trigonometric functions.
Solution. Using the first product-to-sum formula with u = 2x ,v = 3x we have
sin 2x cos 3x =1
2
(sin(2x + 3x) + sin(2x − 3x)
)=
1
2
(sin 5x + sin(−x)
)=
1
2
(sin 5x − sin x
).
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Product-to-Sum Formulas
sin u + sin v = 2 sinu + v
2cos
u − v
2
sin u − sin v = 2 cosu + v
2sin
u − v
2
cos u + cos v = 2 cosu + v
2cos
u − v
2
cos u − cos v = −2 sinu + v
2sin
u − v
2.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Write sin 2x − sin 7x as a product of trigonometricfunctions.
Solution. Using the second sum-to-product formula with u = 2x ,v = 7x we have
sin 2x − sin 7x = 2 cos2x + 7x
2sin
2x − 7x
2
= 2 cos9x
2sin−5x
2
= −2 cos9x
2sin
5x
2.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Write sin 2x − sin 7x as a product of trigonometricfunctions.
Solution. Using the second sum-to-product formula with u = 2x ,v = 7x we have
sin 2x − sin 7x = 2 cos2x + 7x
2sin
2x − 7x
2
= 2 cos9x
2sin−5x
2
= −2 cos9x
2sin
5x
2.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Write sin 2x − sin 7x as a product of trigonometricfunctions.
Solution. Using the second sum-to-product formula with u = 2x ,v = 7x we have
sin 2x − sin 7x = 2 cos2x + 7x
2sin
2x − 7x
2
= 2 cos9x
2sin−5x
2
= −2 cos9x
2sin
5x
2.
MATH 201 - Week 11
Double-Angle, Half-Angle and Product-Sum Formulas - Section 7.3
Product-Sum Formulas
Example. Write sin 2x − sin 7x as a product of trigonometricfunctions.
Solution. Using the second sum-to-product formula with u = 2x ,v = 7x we have
sin 2x − sin 7x = 2 cos2x + 7x
2sin
2x − 7x
2
= 2 cos9x
2sin−5x
2
= −2 cos9x
2sin
5x
2.
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