Angles-
The amount of rotation about the point of intersection of two planes (or lines) which is required to bring one in correspondence with the other is called an Angle.
Representation-
It is generally represented by a Greek letters such as \(\theta, \alpha , \beta\) etc.
It can also be represented by three letters of the shape that define the angle, with the middle letter being where the angle actually is (i.e.its vertex).
Eg. \(\angle ABC\), where B is the given angle.
Angles are measured in terms of degree (\(^{\circ}\)), radians or gradians.
Types of Angles-
Type of angles |
Description |
Acute Angle |
An Angle less than \(90^{\circ}\) |
Obtuse Angle |
An Angle greater than \(90^{\circ}\). |
Right Angle |
An Angle equal to \(90^{\circ}\). |
Straight Angle |
An Angle which is exactly \(180^{\circ}\). |
Reflex Angle |
An Angle greater than \(180^{\circ}\). |
Range of Angles-
Acute Angle – \(0^{\circ}\) to \(90^{\circ}\), both exclusive.
Obtuse Angle – \(90^{\circ}\) to \(180^{\circ}\), both exclusive.
Right Angle – Exactly \(90^{\circ}\).
Straight Angle – Exactly \(180^{\circ}\).
Reflex Angle – \(180^{\circ}\) to \(360^{\circ}\), both exclusive.
Full Rotation – Exactly \(360^{\circ}\)
Concept of Positive & Negative Angles-
Positive Angle- An Angle measured in an Anti-Clockwise direction is Positive Angle.
Negative Angle- An angle measured in Clockwise direction is Negative Angle.
Parts Required to define an Angle-
Vertex- The corner points of an angle is known as Vertex. It is the point where two rays meet.
Initial Side – It is also known as a reference line. All the measurements are done taking this line as the reference.
Terminal Side- It is the side (or ray) up to which the angle measurement is done.
Angle Measurement-
To measure everything in this world, we need a unit in a similarly angle measurement requires three units of measurement-
Degree: It is represented by ° (read as degree). It most likely comes from Babylonians, who used a base 60 (Sexagesimal) number system. In their calendar, there were total 360 days. Hence, they adopted a full angle to be 360°. First, they tried to divide a full angle into angles using angle of an equilateral triangle. Later, following their number system (base 60), they divided 60° by 60 and defined that as 1°. Sometimes, it is also referred as arc degree or arc-degree which means degree of an arc.
Definition: An angle is said to be equal to 1° if the rotation from the initial to terminal side is equal to \( \frac {1}{360} \) of the full rotation.
A degree is further divided into minutes and seconds. 1′ (1 minute) is defined as one sixtieth of a degree and 1” (1 second) is defined as one sixtieth of a minute. Thus,
\(1^{\circ}= 1′\)
\(1′ = 1”\)
Radian: This is SI unit of angle. Radian is mostly used in Calculus. All the formula for derivatives and integrals hold true only when angles are measured in terms of radian. It is denoted by ‘rad’.
Definition: The length of arc of a unit circle is numerically equal to the measurement in radian of the angle that it subtends.
In a complete circle, there are \(2 \pi\) radians.
\(360^{\circ} = 2 \pi \;\; radian \)
Therefore, \(1 \;\; radian = \frac{180^{\circ}}{\pi }\)
Gradian: This unit is least used in Maths. It is also called a gon or a grade.
Definition: An angle is equal to 1 gradian if the rotation from the initial to terminal side is \( \frac {1}{400} \) of the full rotation. Hence, full angle is equal to 400 gradians.
It is denoted by ‘grad’.
Figure 3 shows the example of angles in gradian.
Figure 3:
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