Download Multiple-Angle and Product-to-Sum Formulas

Multiple-Angle and Product-to-Sum Formulas
MATH 160, Precalculus
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Objectives
In this lesson we will learn to:
use multiple-angle formulas to rewrite and evaluate
trigonometric functions,
use power-reducing formulas to rewrite and evaluate
trigonometric functions,
use half-angle formulas to rewrite and evaluate
trigonometric functions,
use product-to-sum and sum-to-product formulas to rewrite
and evaluate trigonometric functions,
use trigonometric formulas to rewrite real-life models.
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Background
Recall the sum and difference of angle formulas we have
learned previously:
sin(u + v ) = sin u cos v + cos u sin v
sin(u − v ) = sin u cos v − cos u sin v
cos(u + v ) = cos u cos v − sin u sin v
cos(u − v ) = cos u cos v + sin u sin v
tan u + tan v
tan(u + v ) =
1 − tan u tan v
tan u − tan v
tan(u − v ) =
1 + tan u tan v
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Double-Angle Formulas
A double-angle is generally expressed as 2u. Thus the
double-angle formulas are:
sin 2u = 2 sin u cos u
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Double-Angle Formulas
A double-angle is generally expressed as 2u. Thus the
double-angle formulas are:
sin 2u = 2 sin u cos u
cos 2u = cos2 u − sin2 u
2 tan u
tan 2u =
1 − tan2 u
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Find the solutions to the following equation in the interval
[0, 2π).
sin 2x + cos x
J. Robert Buchanan
= 0
Multiple-Angle and Product-to-Sum Formulas
Example
Find the solutions to the following equation in the interval
[0, 2π).
sin 2x + cos x
= 0
2 sin x cos x + cos x
= 0
cos x(2 sin x + 1) = 0
1
2
π 3π 7π 11π
,
,
,
2 2 6
6
cos x = 0 or sin x = −
x
J. Robert Buchanan
=
Multiple-Angle and Product-to-Sum Formulas
Example
Find the exact values of sin 2u, cos 2u, and tan 2u given that
cot u = −5 and 3π/2 < u < 2π.
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Find the exact values of sin 2u, cos 2u, and tan 2u given that
cot u = −5 and 3π/2 < u < 2π.
Using right triangle trigonometry we see that
√
√
− 26
5 26
1
sin u =
,
cos u =
,
tan u = − .
26
26
5
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Find the exact values of sin 2u, cos 2u, and tan 2u given that
cot u = −5 and 3π/2 < u < 2π.
Using right triangle trigonometry we see that
√
√
− 26
5 26
1
sin u =
,
cos u =
,
tan u = − .
26
26
5
√
26
sin 2u = 2 sin u cos u = 2 −
26
!
√ !
5 26
5
=−
26
13
25
1
12
−
=
26 26
13
−2/5
2 tan u
5
=
=−
1 − 1/25
12
1 − tan2 u
cos 2u = cos2 u − sin2 u =
tan 2u =
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Verify the following identity.
(sin x + cos x)2 = 1 + sin 2x
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Verify the following identity.
(sin x + cos x)2 = 1 + sin 2x
sin2 x + 2 sin x cos x + cos2 x
2
=
2
(sin x + cos x) + (2 sin x cos x) =
1 + sin 2x
J. Robert Buchanan
=
Multiple-Angle and Product-to-Sum Formulas
Power-Reducing Formulas
Power-reducing formulas replace powers of trigonometric
functions with simpler expressions.
Power-Reducing Formulas
sin2 u =
cos2 u =
tan2 u =
J. Robert Buchanan
1 − cos 2u
2
1 + cos 2u
2
1 − cos 2u
1 + cos 2u
Multiple-Angle and Product-to-Sum Formulas
Example
Use power-reducing formulas to rewrite the following
expression in terms of the first power of cosine.
sin2 x cos2 x
=
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use power-reducing formulas to rewrite the following
expression in terms of the first power of cosine.
1 + cos 2x
1 − cos 2x
2
2
sin x cos x =
2
2
1
=
(1 − cos 2x)(1 + cos 2x)
4
1
=
(1 − cos2 2x)
4
1
=
sin2 2x
4
1 1 − cos 4x
=
4
2
1
=
(1 − cos 4x)
8
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Half-Angle Formulas
Closely related to the power-reducing formulas are the
half-angle formulas which are obtained by replacing u by u/2.
Half-Angle Formulas
u
sin
2
u
cos
2
u
tan
2
r
1 − cos u
2
1 + cos u
= ±
2
1 − cos u
sin u
=
=
sin u
1 + cos u
= ±
r
u
u
The signs of sin and cos depend on the quadrant in which
2
2
u
falls.
2
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use the half-angle formulas to determine the exact values of
sin 165◦ =
cos 165◦ =
tan 165◦ =
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use the half-angle formulas to determine the exact values of
s
√
◦ r
◦
1 − cos 330
1 − 3/2
330
◦
sin 165 = sin
=
=
2
2
2
cos 165◦ =
tan 165◦ =
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use the half-angle formulas to determine the exact values of
s
√
◦ r
◦
1 − cos 330
1 − 3/2
330
◦
sin 165 = sin
=
=
2
2
2
s
r
√
◦
330
1 + cos 330◦
1 + 3/2
◦
cos 165 = cos
=−
=−
2
2
2
tan 165◦ =
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use the half-angle formulas to determine the exact values of
s
√
◦ r
◦
1 − cos 330
1 − 3/2
330
◦
sin 165 = sin
=
=
2
2
2
s
r
√
◦
330
1 + cos 330◦
1 + 3/2
◦
cos 165 = cos
=−
=−
2
2
2
s
s
√
√
1 − 3/2
2− 3
√
√
=−
tan 165◦ = −
1 + 3/2
2+ 3
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Product-to-Sum Formulas
If we combine the sum and difference of angles formulas we
can derive the product-to-sum formulas.
Product-to-Sum Formulas
sin u sin v
=
cos u cos v
=
sin u cos v
=
cos u sin v
=
1
[cos(u − v ) − cos(u + v )]
2
1
[cos(u − v ) + cos(u + v )]
2
1
[sin(u + v ) + sin(u − v )]
2
1
[sin(u + v ) − sin(u − v )]
2
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use the product-to-sum formulas to write the following as a
sum or difference.
sin
π
π
cos
4
12
=
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use the product-to-sum formulas to write the following as a
sum or difference.
π
π
π
1 h π
π
π i
sin cos
=
sin
+
+ sin
−
4
12
2
4 12
4 12
π i
1 h π =
sin
+ sin
2
3
6
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use the product-to-sum formulas to write the following as a
sum or difference.
π
π
π
1 h π
π
π i
sin cos
=
sin
+
+ sin
−
4
12
2
4 12
4 12
π i
1 h π =
sin
+ sin
2"
3 #
6
√
1
3 1
+
=
2 2
2
√
3+1
=
4
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Sum-to-Product Formulas
Occasionally we may wish to rewrite a sum or difference of
trigonometric functions as a product.
Sum-to-Product Formulas
u−v
u+v
cos
2 sin
2
2
u+v
u−v
2 cos
cos
2
2
u−v
u+v
2 cos
cos
2
2
u+v
u−v
−2 sin
sin
2
2
sin u + sin v
=
sin u − sin v
=
cos u + cos v
=
cos u − cos v
=
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use the sum-to-product formulas to find the exact value of the
following expression.
cos 120◦ + cos 60◦ =
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Example
Use the sum-to-product formulas to find the exact value of the
following expression.
120◦ + 60◦
120◦ − 60◦
cos 120◦ + cos 60◦ = 2 cos
cos
2
2
◦
◦
= 2 cos (90 ) cos (30 )
= 0
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Application
The range of a projectile fired at an angle θ with respect to the
horizontal and with an initial velocity of v0 feet per second is
r=
1 2
v sin 2θ.
32 0
If an athlete throws a javelin at 75 feet per second, at what
angle must the javelin be thrown so that it travels 130 feet?
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Solution
1
(75)2 sin 2θ
32
4160 = 5625 sin 2θ
130 =
0.739556 = sin 2θ
2θ = arcsin(0.739556) = 0.83241 ≈ 47.69◦
θ ≈ 23.85◦
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas
Homework
Read Section 5.5.
Exercises: 1, 5, 9, 13, . . . , 133, 137
J. Robert Buchanan
Multiple-Angle and Product-to-Sum Formulas