Measure of Angles in Trigonometry
The
concept of measure of angles in trigonometry is more general compared to a
geometrical angle.
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°.
A angle θ is said to be acute angle if 0° ≤ θ < 90°
A angle θ is said to be right angle if θ = 90°
A angle θ is said to be obtuse angle if 90° < θ < 180°
A angle θ is said to be straight angle if θ = 180°
A angle θ is said to be reflex angle if 180° < θ < 360°
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.
Let
OX be a fixed line on the plane of this page and OA be a revolving line whose initial position coincides with
OX. If
OA begins to revolve about O and comes from its initial position
OX to the final position
OA then we say that
OA forms < XOA with
OX. Here, ∠XOA is called a
trigonometrical angle, O is its vertex,
OX the initial arm and
OA the final arm of the angle. If
OA revolves about O in the anti-clockwise sense and starting from the initial position
OX comes to the final position OA then ∠XOA = (θ) formed by the generating line
OA is called a
trigonometrical positive angle. Conversely, if the generating line
OA revolves about O in the clockwise sense and starting from the initial position
OX comes to the position
OA then ∠XOA (=α) formed by
OA is called a
trigonometrical negative angle.
A trigonometrical angle may have any positive or negative value i.e., such an angle has no definite limit. To make the point clear we take a fixed point O on the plane of the paper and draw two mutually perpendicular lines
XOX’ and
YOY’ through O.
Clearly, the drawn two lines divide the plane of the paper into four regions XOY , YOX‘, X ‘OY‘ and Y‘OX ; these four regions are respectively called the
first,
second,
third and
fourth quadrants. Now, assume that the generating line
OA revolves about O in the anti-clockwise sense and starting from the initial position
OX comes in the positions
OA,
OB,
OC,
OD describing angles ∠XOA, ∠XOB, ∠XOC and ∠XOD in the first, second, third and fourth quadrants respectively.
Clearly, each of the angles ∠XOA, ∠XOB, ∠XOC, ∠XOD is positive and 0 < ∠XOA < 90°, 90° < ∠XOB < 180°, 180° < ∠XOC < 270° and 270° < ∠XOD < 360°.
Thus, any positive angle between 0° and 360° can be described by the revolving line if it does not complete a complete revolution in the anti-clockwise sense and the angle 360° is described when it coincides with
OX after a complete revolution. If
OA revolves further in the same direction then an angle greater than 360° is described by it. Clearly, an angle between 360° and 720° is described by the revolving line
OA if it completes one revolution but does not complete two revolutions in the anti-clockwise sense. In this way a positive angle of any given magnitude can be described by
OA by its repeated revolution in the anti-clockwise sense.
For example, consider the measure of angles in trigonometry 2770°. Since 2770° = 7 × 360° + 180° + 70°, hence, angle of magnitude 2770° is described by the revolving line
OA if it coincides with
OC in the third quadrant after making seven complete revolutions in the anti-clockwise sense. Similarly, if the revolving line
OA starts from the initial position
OX and revolves about O in the clockwise sense, then negative angle of any given magnitude can be described by
OA.
● Measurement of Angles
11 and 12 Grade Math
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