Positive and Negative Angles

The topic of positive and negative angles form an important part and the basics of trigonometry. The word trigonometry itself originates from the Greek words ‘trigōnon’ and ‘metron’ which means triangle and measure, respectively. Hence, trigonometry is a subject that studies the measurement of angles and different functions surrounding it. The application of these functions, also called trigonometric functions, are used in measuring the distance between stars, the height of mountains, construction of houses, roads and bridges, in navigation and many other amazing things. And all of this starts with an angle. This article will cover the positive and negative angles notes. 

What are angles?

In geometry, an angle is formed as a result of the intersection or point of contact of 2 rays. It can also be understood as the degree of rotation of a ray from its initial point. In the latter concept, the best way to understand is to imagine the hands of a wall-clock. The hands of a clock always form an angle with each other. 

The point where the two hands meet is called a vertex. The original position of the small hand is called the initial side of the angle. The position of the small hand after rotation is called the terminal side of the angle.

Positive and negative angles

The positivity or negativity of the angles depends on the direction of the rotation of the ray. Positive and negative angles are formed as a result of the ray in an anticlockwise or clockwise direction.

On a two-dimensional plane, if a ray rotates in an anticlockwise direction, then the angle formed as a product of the rotation is called a positive angle. A positive angle is denoted by θ. 

On the other hand if a ray rotates in a clockwise direction, it forms a negative angle. A negative angle is denoted by -θ. 

Degree and radian measure of angles

In geometry, there are two ways to measure angles – degrees and radians. Radians form the standard unit of measuring an angle. The conversion from radians to degrees and vice versa is an important concept in the basics of trigonometry.

Degree or sexagesimal measure of angles

When a ray makes a rotation from the initial side to reach the terminal side as such that both collide with each other then the angle measures 360 degrees. In simple terms, when an arc makes one full revolution of the minute hand from 12 and back, it is said to form an angle of 360°. Thus, 1 degree is 1360 of the revolution. 

• 1 degree is divided into 60 minutes or 1° = 60’.
• Each minute is divided into 60 seconds. Hence, 1’ = 60’’

Radian or circular measure of angles

Radians are considered as the SI unit to measure angles. When a ray of length 1 unit rotates to form an arc of length 1 unit, then the angle so formed is said to measure 1 radian. Since the circumference of a circle is 2π, one complete rotation of an arc forms an angle of 2π radians. 

Relation between degree and radian

In simple terms, 360° = 2π radians. Thus, 180° = π radians. From the value of pi being 227, it can be concluded that 1 degree = 0.01746 radians. 

Coterminal angles

As mentioned earlier, the various parts of an angle include an initial side, a terminal side and a vertex. Coterminal angles share the same terminal side, and hence the name. There are two types of coterminal angles, positive coterminal and negative coterminal angles. In order to find a positive coterminal of an angle, simply add 360° in the given angle. In order to find the negative coterminal, subtract 360° from the given angle. If the angle is given in radians, simply use 2π in the place of 360°. 

Positive and negative angles questions

Here are some positive and negative angles questions.

Conclusion

Angles are formed when two rays intersect or have a point of contact. Positive and negative angles are determined on the basis of the direction in which the ray rotates. If it rotates in a clockwise direction then the angle is negative and if it rotates in an anticlockwise direction then the angle is positive. There are two ways of measuring angles – the degree method and the radian method. Coterminal angles share the same terminal side.