I can say that negative angle usage is present in trigonometry (vector, displacement, 3D geometry) and not in plane geometry.
Similarly, exterior angle terminology (reflex angle) or concept in rays contradicts the usage in shapes.
Below I have discussed to support my above statement.
Exterior Angle
The larger part of an angle. Were one of the rays of an angle to be rotated until it met the other ray, an exterior angle is spanned by the greater rotation of the two possible rotations. The measure of an exterior angle is always greater than 180 degrees and is always 360 degrees minus the measure of the interior angle that accompanies it. Together, an interior and exterior angle span the entire plane.
Interior Angle
The smaller part of an angle, spanned by the space between the rays that form an angle. Its measure is always less than 180 degrees, and is equal to 360 degrees minus the measure of the exterior angle.
Let us recall about the ray. It has only one end point. The arrow shows that it is extended indefinitely in one direction.
When two rays meet at a point, they form an angle. If OA and OB are rays. They meet at a common point O and form an angle named AOB. An angle is represented by the symbol (∠AOB).
The inside space between the arms in known as the interior of the angle. The outside space is called the exterior of the angle.
We name the angles by using any point on one ray, the vertex and ray point on the other ray. We name an angle with the vertex in the middle. For example, the above mentioned angle is represented by ∠AOB. Thus, When two sides or line-segments are inclined and meet at a point, then this inclination is called an angle.
Reflex angles are angles measuring greater than 180 degrees and less than 360 degrees. The measure of a reflex angle is added to an acute or obtuse angle to make a full 360 degree circle.
Exterior and interior angles in shapes:
The Exterior Angle is the angle between any side of a shape, and a line extended from the next side.
When we add up the Interior Angle and Exterior Angle we get a straight line 180°. They are "Supplementary Angles".
Interior vs. Exterior Angles in shapes
Interior angles are the angles inside the triangle. Exterior angles are the angles made by the outside of the triangle and an extension of one of its sides. Every triangle has three interior angles and six possible exterior angles.
Geometrical angles are always positive. In other words in geometry there is no use of negative angles. But the measure of angles in trigonometry is formed by the revolution of a straight line about a fixed point and the magnitude of such angle has no definite limit i.e., a trigonometrical angle may have any positive or negative value.
More than thousands of years ago, the ancient Babylonians chose 360 as their number to measure angles. An angle in geometry is supposed to be formed by the intersection of two lines and always varies from 0 to 360°. The unit of an angle is called a ‘degree’ (°). One full rotation indicates 360°.
Angle is a measurement and can't be negative, and it is used to define a certain point on the trigonometric circle. In the trigonometric circle, counterclockwise rotation is denoted the positive sign and clockwise is denoted the negative sign. for example, moving 60 degrees in the positive direction is completely identical to moving 300 degrees in the negative direction, as they both define one point. Note that the choice of + or - is completely arbitrary and makes no difference; what really matters is the absolute value of angular displacement.
Negative angles has to do with the direction of rotation that you consider in order to measure angles.
When you have to use words like east west left or right. negative is the equivalent of these words in math.
I can walk positive 10 feet or negative 10 feet along the x axis. Same distance, different directions. a positive angle starts from an initial side and moves clockwise to its terminal side. A negative angle starts from an initial side and moves counterclockwise to its terminal side.
Negative angles are a way of measuring an angle from a different direction. A positive angle is created by rotating counter-clockwise around the origin of a coordinate system, starting at the x-axis (the horizontal axis) and proceeding through the quadrants in the order I, II, III, IV. A negative angle is measured in the clockwise direction from the positive horizontal axis, and proceeds through the quadrants in the order IV, III, II, I. Every negative angle has a positive counterpart. For example, the negative angle -60° leads to the same terminus as the positive angle 300°, because proceeding 60° clockwise is the same as proceeding 300o counter-clockwise. Those two angles are coterminal, meaning that they terminate at the same location.
If the angle between A and B are greater than 90 degrees, the dot product will be negative (less than zero), as cos(Θ) will be negative, and the vector lengths are always positive values
Angles is a space between two intersecting lines and is measured in degree(in case of plane) or radians(in case of solid).
Plane angle- usually refers to an angle in 2 dimensional figure like square, triangle, circle etc
Solid angle-usually refers to an angle in 3 dimensional figure like cube, cuboid, sphere etc
The concept of measure of angles in trigonometry is more general compared to a geometrical angle.
Thus I like to conclude that there are different systems which have different usage or limited usage of the terminology or the usage of concept restricted to certain application only.
Similarly, exterior angle terminology (reflex angle) or concept in rays contradicts the usage in shapes.
Below I have discussed to support my above statement.
Exterior Angle
The larger part of an angle. Were one of the rays of an angle to be rotated until it met the other ray, an exterior angle is spanned by the greater rotation of the two possible rotations. The measure of an exterior angle is always greater than 180 degrees and is always 360 degrees minus the measure of the interior angle that accompanies it. Together, an interior and exterior angle span the entire plane.
Interior Angle
The smaller part of an angle, spanned by the space between the rays that form an angle. Its measure is always less than 180 degrees, and is equal to 360 degrees minus the measure of the exterior angle.
Let us recall about the ray. It has only one end point. The arrow shows that it is extended indefinitely in one direction.
When two rays meet at a point, they form an angle. If OA and OB are rays. They meet at a common point O and form an angle named AOB. An angle is represented by the symbol (∠AOB).
The inside space between the arms in known as the interior of the angle. The outside space is called the exterior of the angle.
We name the angles by using any point on one ray, the vertex and ray point on the other ray. We name an angle with the vertex in the middle. For example, the above mentioned angle is represented by ∠AOB. Thus, When two sides or line-segments are inclined and meet at a point, then this inclination is called an angle.
Reflex angles are angles measuring greater than 180 degrees and less than 360 degrees. The measure of a reflex angle is added to an acute or obtuse angle to make a full 360 degree circle.
Exterior and interior angles in shapes:
The Exterior Angle is the angle between any side of a shape, and a line extended from the next side.
When we add up the Interior Angle and Exterior Angle we get a straight line 180°. They are "Supplementary Angles".
Interior vs. Exterior Angles in shapes
Interior angles are the angles inside the triangle. Exterior angles are the angles made by the outside of the triangle and an extension of one of its sides. Every triangle has three interior angles and six possible exterior angles.
Geometrical angles are always positive. In other words in geometry there is no use of negative angles. But the measure of angles in trigonometry is formed by the revolution of a straight line about a fixed point and the magnitude of such angle has no definite limit i.e., a trigonometrical angle may have any positive or negative value.
More than thousands of years ago, the ancient Babylonians chose 360 as their number to measure angles. An angle in geometry is supposed to be formed by the intersection of two lines and always varies from 0 to 360°. The unit of an angle is called a ‘degree’ (°). One full rotation indicates 360°.
Angle is a measurement and can't be negative, and it is used to define a certain point on the trigonometric circle. In the trigonometric circle, counterclockwise rotation is denoted the positive sign and clockwise is denoted the negative sign. for example, moving 60 degrees in the positive direction is completely identical to moving 300 degrees in the negative direction, as they both define one point. Note that the choice of + or - is completely arbitrary and makes no difference; what really matters is the absolute value of angular displacement.
Negative angles has to do with the direction of rotation that you consider in order to measure angles.
When you have to use words like east west left or right. negative is the equivalent of these words in math.
I can walk positive 10 feet or negative 10 feet along the x axis. Same distance, different directions. a positive angle starts from an initial side and moves clockwise to its terminal side. A negative angle starts from an initial side and moves counterclockwise to its terminal side.
Negative angles are a way of measuring an angle from a different direction. A positive angle is created by rotating counter-clockwise around the origin of a coordinate system, starting at the x-axis (the horizontal axis) and proceeding through the quadrants in the order I, II, III, IV. A negative angle is measured in the clockwise direction from the positive horizontal axis, and proceeds through the quadrants in the order IV, III, II, I. Every negative angle has a positive counterpart. For example, the negative angle -60° leads to the same terminus as the positive angle 300°, because proceeding 60° clockwise is the same as proceeding 300o counter-clockwise. Those two angles are coterminal, meaning that they terminate at the same location.
If the angle between A and B are greater than 90 degrees, the dot product will be negative (less than zero), as cos(Θ) will be negative, and the vector lengths are always positive values
Angles is a space between two intersecting lines and is measured in degree(in case of plane) or radians(in case of solid).
Plane angle- usually refers to an angle in 2 dimensional figure like square, triangle, circle etc
Solid angle-usually refers to an angle in 3 dimensional figure like cube, cuboid, sphere etc
The concept of measure of angles in trigonometry is more general compared to a geometrical angle.
Thus I like to conclude that there are different systems which have different usage or limited usage of the terminology or the usage of concept restricted to certain application only.
No comments:
Post a Comment