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.
sinθ is a constructible number;
-
3.
cosθ is a constructible number.
Proof.
First of all, due to periodicity, we can restrict our attention to the interval 0≤θ<2π. Even better, we can further restrict our attention to the interval 0≤θ≤π2 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 2π-θ;
-
2.
If x is a constructible number, then so is |x|.
If θ∈{0,π2}, then clearly an angle of measure θ is constructible, and {sinθ,cosθ}={0,1}. Thus, equivalence (http://planetmath.org/Equivalent3) has been established in the case that θ∈{0,π2}. Therefore, we can restrict our attention even further to the interval 0<θ<π2.
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 sinθ and cosθ, both of these are constructible numbers.
Now assume that sinθ is a constructible number. At one endpoint of a line segment of length sinθ, 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 cosθ have been constructed.
A similar procedure can be used given that cosθ is a constructible number to prove the other two statements.
∎
Note that, if cosθ≠0, then any of the three statements thus implies that tanθ is a constructible number. Moreover, if tanθ is constructible, then a right triangle having a leg of length 1 and another leg of length tanθ 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 |