Measurement of Angle

Understanding trigonometry, a study initially meant to solve geometric issues involving triangles, requires a thorough knowledge of the measurement of angles. Understanding the distinct methods of measurement of angles will help to have an idea about the topic. For example, the movement of a ray from a beginning point to a conclusion point is known as an angle. The degree of rotation done by the beginning side to get to the lateral edge is also known as the measurement of angle. 

What is the Measurement of Angle?

The measurement of angle shall be done in a variety of ways. Angles are measured in degrees, radians, and revolutions. Although radians are perhaps the most commonly used measurement in trigonometry, it is critical to be able to transform between all three units. For example, a full rotation is split into 360 degrees. It means when the beginning and final edges are from the same spot after moving clockwise or anticlockwise. The angle seems to be positive if the movement is clockwise. And the angle appears to be negative if the movement is in an anticlockwise direction.

How does Angle Measurement work?

The measurement of angle shall be done in three distinct ways. They are 

• Degree measure
• Angle measure
• Grade measure

Degree Measure in Measurement of Angle:

A degree is a complete revolution in either a counterclockwise or clockwise direction with identical starting and ending points. The 360-degree movement is broken into units.

It shall be one degree if the movement from the starting and finishing sides is 1/360th of the rotation. 

1/360th of the rotation = 1°

Time shall be recorded in seconds, minutes, and hours. So, one hour shall equal 60 minutes, and 60 seconds shall equal one minute. We shall follow the same order in the measurement of angles

60 minutes =  1 degree. We can represent it as 60′ = 1°

60 seconds = 1 minute. We can represent it as  60° = 1′

Radian Measure in Measurement of Angle:

The radian scale is a little more complicated than the degree scale. Consider a circular radius of one unit. Next, visualise a one-unit-long arc of the circle. The angle occupied by this arc at the circle’s centre equals 1 radian.

The circle’s arc length of a circle with radius unity or one is equal to the angle in radian.

r * = l

(l / r ) =

r is the circle radius

l is the arc length

is radian

Grade Measure in Measurement of Angle:

A grade is a straight angle split into a hundred equal sections. In addition, each grade is made up of 100 minutes, with each minute being broken into 100 seconds.

How the Relationship Between Radian & Degree Measures Works?

According to the standards of radian and degree measurements, the angle surrounded by a circle at its centre is 360° and two π as per the degree measure.

360° (degrees)  = 2 * π (radians)

(180° / π ) = 1 radian

Hence 1 radian shall be calculated using the above formula.

180° (degrees) = π (radians)

(π  / 180 ) = 1 degree

 

Hence 1 degree shall be calculated using the above formula.

These are the relationships between grade and degree measures.

Measurement of Angle Notes:

The critical measurement of angle notes are as follows:

• A degree or 1° is a fraction of a whole rotation.
• If the vertex or edge of an angle is at zero and one arc is on the positive x-axis, it would be in the standard position. The arc on the x-axis is known as the starting side, while the other ray is known as the lower edge.
• Angles more significant than 180 degrees or (180°) but less than 360 degrees or (360°) are known as reflex angles. To construct a full (360°) circle, we add the measurement of a reflex angle to an obtuse angle or acute angle.

Measurement of Angle Important Problems:

Some of the measurement of angle important problems are as follows:

• What seems to be the triangle inner angles sum?

Since each triangle side equals 60°, its inner angles total is 180° (degrees)

• A triangle’s interior angles sum up to 52°. What is the measurement of another angle?

The triangle’s three angles sum equals 180°. To find the third angle, subtract 52 degrees.

180° – 52° = 128°.

• A triangle’s interior angles are 50° and 45°. So which of its exterior angles has the most extensive measurement?

A triangle’s inner angles are (60° + 60° + 60°) 180°. Hence the third angle’s measure must be  

3rd angle measure = (180) – (50 + 45). Hence, it’s 85°.

According to the Triangle Exterior-Angle Theorem, a triangle’s exterior angle measures the sum of its distant interior angles. So, to acquire the full measure of any exterior angle, we combine the two biggest interior measurements of angle:

(50 + 85) = 135°.

Conclusion:

Mastering trigonometry, which was developed to address geometric problems involving triangles, demands a good understanding of angle measurement. Knowing the different techniques of measuring angles can help you better understand the subject. An angle is defined as the distance a beam travels from its origin to its destination. Angle measurement refers to the degree of rotation performed by the starting side to reach the lateral edge. Angles may be measured in several different ways. Degrees, radians, and revolutions are all used to measure angles. Although radians are the most widely used unit in trigonometry, the ability to convert between all three units is essential.