theorem on constructible angles

theorem on constructible angles

Theorem 1

.

Let

θ

∈

R

. Then the following are

equivalent

:

1.

An angle of measure (

http://planetmath.org/AngleMeasure

)

θ

is

constructible

(

http://planetmath.org/Constructible2

);

2.

sin

⁡

θ

is a constructible number;

3.

cos

⁡

θ

is a constructible number.

Proof.

First of all, due to periodicity, we can restrict our attention to the

interval

0

≤

θ

<

2

⁢

π

. Even better, we can further restrict our attention to the interval

0

≤

θ

≤

π

2

for the following reasons:

1.

If an angle whose measure is

θ

is constructible, then so are angles whose measures are

π

-

θ

,

π

+

θ

, and

2

⁢

π

-

θ

;

2.

If

x

is a constructible number, then so is

|

x

|

.

If

θ

∈

{

0

,

π

2

}

, then clearly an angle of measure

θ

is constructible, and

{

sin

⁡

θ

,

cos

⁡

θ

}

=

{

0

,

1

}

. Thus, equivalence (

http://planetmath.org/Equivalent3

) has been established in the case that

θ

∈

{

0

,

π

2

}

. Therefore, we can restrict our attention even further to the interval

0

<

θ

<

π

2

.

Assume that an angle of measure

θ

is constructible. Construct such an angle and mark off a

line segment

of length

1

from the vertex (

http://planetmath.org/Vertex5

) of the angle. Label the

endpoint

that is not the vertex of the angle as

A

.

Drop the

perpendicular

from

A

to the other ray of the angle. Since the legs of the

triangle

are of lengths

sin

⁡

θ

and

cos

⁡

θ

, both of these are constructible numbers.

Now assume that

sin

⁡

θ

is a constructible number. At one endpoint of a line segment of length

sin

⁡

θ

, erect the perpendicular to the line segment.

From the other endpoint of the given line segment, draw an arc of a circle with radius

1

so that it

intersects

the erected perpendicular. Label this point of

intersection

as

A

. Connect

A

to the endpoint of the line segment which was used to draw the arc. Then an angle of measure

θ

and a line segment of length

cos

⁡

θ

have been constructed.

A

similar

procedure can be used given that

cos

⁡

θ

is a constructible number to prove the other two statements.
∎

Note that, if

cos

⁡

θ

≠

0

, then any of the three statements thus implies that

tan

⁡

θ

is a constructible number. Moreover, if

tan

⁡

θ

is constructible, then a

right triangle

having a leg of length

1

and another leg of length

tan

⁡

θ

is constructible, which implies that the three listed conditions are true.

Title

theorem on constructible angles

Canonical name

TheoremOnConstructibleAngles

Date of creation

2013-03-22 17:15:59

Last modified on

2013-03-22 17:15:59

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

13

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 33B10

Classification

msc 51M15

Classification

msc 12D15

Related topic

ConstructibleNumbers

Related topic

CompassAndStraightedgeConstruction

Related topic

ConstructibleAnglesWithIntegerValuesInDegrees

Related topic

ExactTrigonometryTables

Related topic

ClassicalProblemsOfConstructibility

Related topic

CriterionForConstructibilityOfRegularPolygon

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