5.2 Double-angle & power-reduction identities

5.2 Double-angle & power-reduction identities

In the previous section we developed formulas for expressions such as cos( + ). These formulas lead naturally to another set of identities involving double angles and half-angles.

5.2.1 Double angle and power reduction formulas Recall the sum-of-angle identities (equations 9 and 10 from the previous section.)

cos( + ) = cos cos - sin sin

sin( + ) = sin cos + cos sin

If we replace and with the same angle, , these identities describe the sine and cosine of 2, expressing trig functions of a doubled angle in terms of the original.

cos(2) = cos2 - sin2

(16)

sin(2) = 2 sin cos

(17)

The first equation, equation 16, is particularly applicable. It can be used to reduce the power on cosine and sine. For example, suppose we add equation 16 to the Pythagorean Identity (cos2 + sin2 = 1, equation 1 from a previous section). We will get 2 cos2 = 1 + cos 2. After dividing by 2, we obtain an equation for cos2 .

cos2

=

1 (1

+

cos 2)

(18)

2

In a similar manner, we could subtract equation 16 from the Pythagorean identity, getting 2 sin2 = 1 - cos 2. After dividing both sides by 2 we have a formula for sin2 :

sin2

=

1 (1 - cos 2)

(19)

2

These are sometimes called "power reduction formulas" because they allow us reduce the power on one of the trig functions when the power is an even integer.

For example, we can reduce the fourth power on cosine in cos4 x = (cos2 x)2 by substituting for cos2 x and then expanding the expression.

cos4 x

=

1 (

+ cos 2x )2

=

1 (1 +

2 cos 2x

+ cos2 2x).

2

4

Since

cos2 x

=

1 2

(1

+

cos

4x)

we

can

again

reduce

the

power

on

cos2

2x,

writing

cos4

x

=

1 (1 +

2 cos 2x

1 + cos 4x

+(

))

4

2

We may then simplify the last expression by using the common denominator 8 and so write:

cos4 x

=

1 (3 +

4 cos 2x

+ cos 4x).

8

In this way we have reduced the exponents on cosine from a fourth power to a first power at the expense of increasing the angles from x to 2x and 4x.

206

5.2.2 Half-angle formulas

The power reduction formulas can also be interpreted as half-angle formulas. Consider again power

reduction

formulas

18

and

19,

above.

Replace

2

by

and

so

replace

by

1 2

.

Now

our

equations

are

cos2( )

=

1 (1 +

cos )

22

and

sin2( )

=

1 (1

-

cos )

22

Now take square roots of both sides:

1

cos( ) = ? (1 + cos )

(20)

2

2

1

sin( ) = ? (1 - cos )

(21)

2

2

If

we

want

an

equation

for

tan(

2

)

we

divide

equation

21

by

equation

20

to

get

1 - cos

tan( ) = ?

(22)

2

1 + cos

Some worked problems Use half-angle formulas to find the exact values of the following:

1.

cos(

12

).

Solution.

We

view

12

as

half

of

the

angle

=

6

.

So

using

the

half-angle

formula

for

cosine,

with

=

6

,

we

have

cos(

12

)

=

1 + cos(/6) =

2

1 + 3/2 =

2

2+ 3 =

4

2+ 3 .

2

2.

sin(

12

).

Solution.

Again

we

view

12

as

half

of

the

angle

=

/6

and

here

we

use

half-angle

formula

for

sine.

sin(

12

)

=

1 - cos(/6) =

2

1 - 3/2 =

2

2- 3 =

4

2- 3 .

2

3.

cos(

24

).

Solution. The

sine

of

12

so

we

angle

24

can use

is

half

of

the

angle

=

12

.

Fortunately

the half-angle formula again:

we

just

computed

the

cosine

and

2+ 3

cos( ) =

1

+

cos(

12

)

=

1+

2

=

24

2

2

2+ 2+ 3

. 2

4.

sin(

24

).

Solution.

sin(

24

)

=

1

-

cos(

12

)

=

2

2+ 3 1-

2= 2

2- 2+ 3

. 2

207

An interesting question.

How

many

square

roots

signs

are

there

in

the

formula

for

sin

48

?

Is

there

a

pattern

to

the

formula

for

sine

and

cosine

if

we

start

at

6

and

keep

cutting

the

angle

in

half? Since the sine function helps give us a chord on the circle, we could then develop a formula for the

circumference of a circle and then develop, from that, a formula for .

5.2.3 Other resources on double angle and power reduction identities

In the free textbook, Precalculus, by Stitz and Zeager (version 3, July 2011, available at stitz-) this material is covered in sections 11.2 and 11.3.

In the free textbook, Precalculus, An Investigation of Functions, by Lippman and Rassmussen (edition 1.3, available at ) this material is covered in sections 7.2 and 7.3.

In the textbook by Ratti & McWaters, Precalculus, A Unit Circle Approach, 2nd ed., c. 2014, here at this material appears in section 5.4. Product-to-sum and sum-to-product formulas are covered in section 5.5.

In the textbook by Stewart, Precalculus, Mathematics for Calculus, 6th ed., c. 2012, (here at ) this material appears in section 7.3 (with applications to all trig equations in sections 7.4 and 7.5.)

Here are some online resources:

1. Khan Academy videos on trig identities;

2. Dr. Paul's online math notes include a review of trig formulas and a guide to solving trig equations.

(Remember: the philosophy of this class is "understand; don't memorize!")

Homework. As class homework, complete Worksheet 5.2, Double Angle and Power Reduction Identities

available through the class webpage and Blackboard.

208

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download