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Angles and their Measurements


The angle is a structure formed with the help of two rays by sharing a common endpoint known as vertex of the angle. There are also trig identities which involves the angles and their measurements. The Trigonometric Identities are separated from the triangle properties, which involve both the angles and sides of the triangle. These identities are helpful in resolving the Functions and also when trig Functions need to be simplified.

The major functions are sine, cosine, tangent, cotangent, secant, cosecant, cotangent. The basic angles which we consider while simplifying the functions begin from 00 to 900which we consider most of the time in Right Angle triangle. They are explained below in tabulation form here in the horizontal format values are there and in vertical format the angles are there:

00 300 450 600 900
Sin C 0 ½ √2/2 √3/2 1
Cos C 1 √3/2 √2/2 ½ 0
Tan C 0 1/√3 1 √3

Now rest of the trigonometric ratios can be found with the help of sin, cos and tan functions because the remaining functions i.e. the cosecant ( cosec), secant (sec), cotangent (cot) are reciprocal function cosine (cos), sine (sin), tangent (tan).
Sec C = 1/ cos C,
Cosec C = 1/ sin C,
Cot C = 1/ tan C =cos C / sin C.

These definitions are also referred as the Ratio identities. Considering the Pythagorean trigonometric identity the basic relationship exist between the sin and cosine:
sin2 θ + cos2 θ = 1.
The equation can also be written as sin θ = √(1- cos2 θ) and cos θ = √( 1- sin2 θ
θ (pronounced as theta) is a Greek letter used here to represent the angle.

Coterminal Angles

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An angle is defined as a figure or form. It is any shape formed by two lines or rays connected as a given end Point. The point in an angle where two rays meet is called to be a vertex. Angles are of different types:
ACUTE ANGLE: An angle that is less than 90 degree.
OBTUSE ANGLE: An angle that is more than 90 degree.
RIGHT ANGLE: An angle that is exactly 90 degree.
REFLEX ANGLE: An angle that is greater than 90 degree.
STRAIGHT ANGLE: An angle that is exactly 180 degree.

Another category of angles is coterminal angles. These angles can be defined as the angles in standard Position. That is the angle with initial side on the positive X-axis that has a common terminal side.
For example: 30 degree, -330 degree and 390 degree are all coterminal angle.

To find positive and negative co terminal angles with a given angle, we can add and subtract 360 degree.
For example :
Question: Find a positive and a negative angle coterminal with a 55 degree angle.
Solution: 55 – 360 = -305 degree.
55 + 360= +415 degree.
Hence a -305 degree and 415 degree angle is coterminal with a 55 degree angle.

Importance of coterminal angle :
In Trigonometry we use the Functions of angles like cos , sin, tan. It turns out that angles that are coterminal have the same value for these Functions.
For example:
Sin 30 , sin 390 and sin 330 all have the same value that is 0.5 .

Difference between Coterminal angles and reference angles:
A reference angle is an angle that the terminal side (the Ray forming from the center of your graph) forms with the X-axis. So if you have a 140 degree angle it takes another 400 degrees to reach the X-axis, so 40 degree is the Reference Angle.

A coterminal angle on the other hand, is an angle that shares the same terminal side with the angle. So if you have a 14 degree angle, it is coterminal with a 374 degree angle, because you have made a complete rotation around the Circle, ending up at the same location.
Thus, this was a brief discussion over coterminal angles.

Use the below widget to calculate coterminal angles.

Quadrant Angles

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In mathematics, quadrant term is used for the “quarter". In a coordinate graph as shown below, the Intersection of the two axes creates four regions. These four regions are called quadrants. Usually they are named by Roman Numbers as I, II, III and IV.

In the first quadrant, both coordinates (x and y) are positive. (+,+)
In the second quadrant, x-coordinates are negative and y-coordinates positive. (−,+)
In the third quadrant, both coordinates are negative. (−,−)
In the fourth quadrant, x-coordinates are positive and y-coordinates negative. (+,−)

And the angles lying in these quadrants are called quadrant angles.
1. In first quadrant the angles lie between 00 and 900.
2. In second quadrant the angles lie between 900 and 1800.
3. In third quadrant the angles lie between 1800 and 2700.
4. In fourth quadrant the angles lie between 2700 and 3600.

Let us see the Functions of some of these quadrant angles.

Function of 00
Sin 00 = 0/a = 0
Cos 00 = a/a = 1
tan 00 = 0/a = 0
cot 00 = a/0 = infinite
Sec 00 = a/a = 1
cosec 00 = a/0 = infinite

Function of 900
Sin 900 = a/a = 1
Cos 900 = 0/a = 0
tan 900 = a/0 = infinite
cot 900 = 0/a = 0
Sec 900 = a/0 = infinite
cosec 900 = a/a = 1

Function of 1800
Sin 1800 = 0/a = 0
Cos 1800 = -a/a = -1
tan 1800 = 0/-a = 0
cot 1800 = -a/0 = infinite
Sec 1800 = a/-a = -1
cosec 1800 = a/0 = infinite

Function of 2700
Sin 2700 = -a/a = -1
Cos 2700 = 0/a = 0
tan 2700 = -a/0 = infinite
cot 2700 = 0/-a = 0
Sec 2700 = a/0 = infinite
cosec 2700 = -a/a = -1

Similarly we can calculate Functions of all quadrant angles.

Positive and Negative Angles

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We come across the terms positive and negative angle, when we study about the Triangles in the Trigonometry. Word trigonometry is derived from the tri - go- no, which means the study of the sides and the angles of the three sided figures. So we say that the study of trigonometry is the science in which we study about the three sided figures and the formation of the angles by the three sided figures, from which one of the angle is 90 degrees.
Along with this we study the Relations of the measurements of the sides of the triangles and their relations with the angles. If we form a right angled triangle, we observe that as the measure of the angle is increased, the measure of the perpendicular dropped on the opposite of that angle also increases. This study relates to the trigonometry.
We define trigonometry as the branch of mathematics which we use to measure the angles of the triangle and to solve the word problems related to it. To study about the formation of angles by rotating the one Ray and keeping the other ray fix at zero degree can be said as negative angles or positive angle.
Let us first look at the positive angle. If the initial side is kept constant and the angle is formed by rotating another side is called the terminal side of the angle in the positive direction of the angle formation is called a positive angle. This movement of the angle is in anti clock wise direction.
On the other hand if we have the initial side constant placed at the zero degree and the other side called the terminal side is moved downward then we say that the negative angle is formed. This movement of the angle is in the clock wise direction.

Measure of Angles in Radians

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In mathematics, when two rays make a figure, which is called as an angle. In Geometry mathematics, an angle is a very important aspect. Trigonometry is based on the measurement of an angle. There are two units which are used in measurement of an angle one is degree and other is radian.
Mostly we use the unit degree which is popular than radian. Further degree was divided into minutes and seconds.
Radian was introduced in the later phase. Ratio between length of an arc and radius is known as Radian. It is a unit which is used for measurement of an angle. Let us take an example, if there is a Circle whose arc is 6unit and whose radius is 3unit, then the measurement of that radian will be 2unit. If we know, length of an arc we can easily calculate radian:
radian measurement * times of radius=arc length
For an example if the circle radius is 6 and its arc is 0.6 which means that the length is 0.6 times 6 , which is 3.6.
angle Degree Radian
90 π/2
60 π/3
45 π/4
30 π/6

Measure of Angles in Radians:
We have to follow some steps if we want to convert a degree into a radian.
Step 1: We have to convert degree minutes and seconds into decimal Numbers. Minutes should be divided by 60 and then add this to number of degrees. Let us take an example to better understand it. So 24 degree 30’ which is equals to 24+30/60 = 24.5 degree.
Step 2: If we want to convert a radian into degree just multiply it with the 180 degree and divide it by ‘π’.
Step 3: If we want to find the arc then simply convert an angle into radian and multiply by its radius.
Step 4: If we want to find the angle in radian then divide the angle by its radius.
With the help of above steps we can measure of angle in radian.

Use the below widget to convert degrees into radians.

Reference Angle

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The term angle is mainly used in the Geometry and Trigonometry. When two rays make or form a figure then they also form an angle. Angle is used in measuring the rotation of any Ray with respect to some fix Point. When a ray inclines over a ray then the two rays form an angle. Angles are of various type, here we will discuss reference angle.
An Acute Angle, which is positive and which can stand for any angle, having different measurement is called a reference angle. Any angle which is placed in the x-y plane has a reference angle. Usually reference angles placed between the 0 degree to 90 degree and it is the smallest angle from the terminal side of an angle with x axis.
Reference angle, is always in the standard Position. An angle is in standard position if its vertices are placed on origin. One ray must be placed in the positive x direction. The ray which is placed in positive x direction is initial ray and the other is terminal ray.
So, it can be said that a reference angle is an acute angle. We can find reference angle just by adding or subtracting 180 degree. We have to subtract the degree of an angle until we get the reference angle between 0 degree to 90 degree.
Let us take an example to understand how to get reference angle? Assume that we have an angle of 30 degree and we want to find its reference angle. So we will subtract 30 degrees from 180 degrees. We get the angle value as 150 degree but this does not come in the Domain of 0 to 90 degrees. We again subtract it from 180 degrees. Now we will get the angle as 30 degree which is in 0 to 90 degrees.
So the reference angle of 30 degree is also 30 degree.

Use the below widget to calculate reference angle.