What is Trigonometry?
Trigonometry is the study of the relationships between a triangle’s side lengths and its Angles. It has various applications, such as in Calculus and the description of Light and Sound Waves.
What is an Angle?
An Angle is described as the measurement of the rotation of a ray about its endpoint. This endpoint is called the Vertex of the Angle. The starting position of the ray is called the initial side, while the final position after rotation is called the terminal side. When the Vertex is at the origin and the initial side is on the positive half of the line y = 0, then the Angle is said to be in standard position.
How would one go about measuring these rotations? Would there be more than one way of showing the amount of rotation?
What is a Degree?
A Degree is used as a unit for measuring Angles. It is defined as 1/360th of a rotation about the Vertex in an anti-clockwise direction.
To make it easier, think of a circle as a pie. Divide the pie into 360 equal chunks. One of these chunks would then be equivalent to a Degree of Rotation.
Another, more effective way of measuring Angles is by using Radians as the unit of measurement. A Radian is defined as the quotient of the arc length (of the Angle that it subtends) divided by the radius of the arc. As this does not depend on an arbitrary value (360 Degrees in this case), it is therefore a better way of recording the value of Angles. One Radian is equal to the Angle that is subtended by an arc at the center of the circle that is equal in length to the radius of the circle.
One complete revolution is equal to 2π Radians. Therefore, using this in combination with the fact that a revolution is equal to 360 Degrees, we can calculate the number of Degrees in a Radian.
2π Radians= 360 Degrees
1 Radian=?
1 Radian= 360 Degrees/2π Radians (cross multiplication)
1 Radian= 57.3 Degrees (to three significant figures)
From the above calculation, we can generalize it as follows:
Radians= (Degree)x(π/180) Degree= (Radians)x(180/π)
Example:
Convert 1.5π Radians to Degrees
Degree= (Radians)x(180/π)
(1.5π) x(180/π) = 270 Degrees
Convert 45 Degrees to Radians
Radians= (Degree)x(π/180)
(45 Degrees) x(π/180)= 0.25π Radians
Note:
The “π” symbol is used to accurately describe the value in Radians. It may not be found on every value of Radians that is given.