Relationship of Degree with Minute and Second: An angle is a shape formed by two rays joining a common vertex. In understanding trigonometry, the various ways of measuring angles help to better hold on to the subject. An angle is the amount of rotation of a ray from an initial side to an end side.

An angle is measured usually in degree. In this article, we will study step-by-step interconversion of degrees to degrees, minutes and seconds; different ways (three ways) to measure the size of an angle in detail. Continue reading to find out more about the relationship of degree with minute and second!

What is an Angle?

An angle is a space between its side. To measure an angle, we compare it with another unit. The most common unit o measurement of an angle is sexagesimal degree consisting of \(360\) parts of a full or complete angle. The measurement of this unit is denoted by the symbol “°”.

An angle is the amount of rotation of a ray from an initial side to an end side. Below we have provided some common terms in angles:

1. Initial side: The original ray
2. End or Terminal side: The end position of the ray after rotation
3. Vertex: The point where initial and end sides meet
4. Positive angle: The direction of rotation is anticlockwise
5. Negative angle: The direction of rotation is clockwise

An angle is measured using a protractor. Protractor gives the measure of an angle in degrees. We can observe from the figure below and the angle indicated measures \({100^ \circ }\).

Examples on Angle Measurement

Some examples of angles with their measurement have been shown below.

Right Angle

A right angle is formed when a straight line is divided equally into two parts. Each of these equal parts is called a right angle. A right angle is important in defining the various system of measuring the angles.

Systems of Measuring Angles

There are three systems of units of measurement of trigonometric angles that are mentioned below:

1. Sexagesimal system or English system
2. Centesimal system or French system
3. Circular system

1. Sexagesimal System

In this system of unit measurement, an angle is measured in degrees, minutes and seconds. One complete revolution is when the start and end sides are in the same position after rotating in either clockwise or anticlockwise direction, divided into \(360\) units. That means a complete angle is divided into \(360\) equal parts, and each part is measured using the unit degree. A degree is indicated using the symbol “°”- for example, \({360^ \circ },{90^ \circ }\)

In the sexagesimal system of units, a right angle is divided into \(90\) equal parts, and each part is equal to \(1\) degree. One degree \(\left( {1^\circ } \right)\) is divided into \(60\) equal parts, and each of the parts is called a minute. One minute is denoted by \(1’\).  Further, one minute is sub-divided into \(60\) equal parts. Each of these \(60\) parts is called second. One second is denoted by \(1″.\)

Let us relate write in short, the relation of degree with minute and minute with seconds

\(1\) right angle \(=90\) degrees \(({90^ \circ })\)

\(1\) degree \(\left( {{1^ \circ }} \right) = 60\) minutes \(\left( {60′} \right)\)

\(1\) minute \((1′)=60\) seconds \(\left( {60”} \right)\)

The above relation results in \(90\) degrees \(\left( {{{90}^ \circ }} \right) = 3600\) seconds \(\left( {3600”} \right)\)

2. Centesimal System

In this system of the unit, measurements of the angle are grades, minutes and seconds. In the centesimal system, a right angle is split into \(100\) equal parts, and each part is known as a grade. One grade is denoted by \(1^{g}\) The grade is again divided into \(100\) equal parts, and each part is called a minute. Further minute is sub-divided into \(100\) equal parts, each of which is called seconds. One minute is denoted as \(1’\) and one second is denoted as \(1″. \)

In other words, 

\(1\) right angle \(=100\) grades \(\left( {{{100}^g}} \right)\)

\(1\) grade \(\left(1^{g}\right)=100\) minutes \(\left( {100′} \right)\)

\(1\) minute \((1′)=100\) seconds \(\left( {100”} \right)\)

The “minutes” and “seconds” that we discussed in the sexagesimal system and centesimal system are different. They have no connection.

Relation between the Sexagesimal and Centesimal System of Units

1. The angle of the sexagesimal system to centesimal system: 

\(1\) right angle, \({90^ \circ } = 90 \times 60 = 5400\) sexagesimal minutes \(=5400’\)

\(1\) right angle, \({90^ \circ } = 90 \times 60 \times 60 = 324000\) sexagesimal seconds \(=324000″\)

2. The angle of the centesimal system to sexagesimal system:

\(1\) right angle \(=100×100=10000\) centesimal minutes \(=10000’\)

\(1\) right angle \(=100×100×100=1000000\) centesimal seconds \(=1000000″\)

Since, \(1\) right angle \( = {90^ \circ } = {100^g}\)

Therefore, \({90^ \circ } = {100^g}\)

or, \({1^ \circ } = {\left( {\frac{{10}}{9}} \right)^g}\) and \({1^g} = {\left( {\frac{9}{{10}}} \right)^ \circ }\)

3. Circular System

In the circular system of unit measurement, an angle is measured in radians. In higher mathematics, the circular system is more often used. Radian measure is a little more complicated than the degree measure.

Definition of radian: A radian is defined as an angle subtended by an arc at the centre whose length is equal to the radius of the circle.

Consider a circle with centre \(O.\) Let \(OX=r\) be the radius of the circle

Now, take an arc of length equal to the radius of the circle,\( XY=r\) and join \(OY.\) By definition, \(\angle X O Y=\) one radian.

One radian is expressed as \(1^{c}\)

In general, \(k\) radian is expressed as \(k^{c}\)

Radian Measure of an Angle

The relation between degree measures and circular (radian) measures of some standard angles are given below:

Degrees	Radians
\({0^ \circ }\)	\(0\)
\({30^ \circ }\)	\(\frac{\pi }{6}\)
\({45^ \circ }\)	\(\frac{\pi }{4}\)
\({60^ \circ }\)	\(\frac{\pi }{3}\)
\({90^ \circ }\)	\(\frac{\pi }{2}\)
\({120^ \circ }\)	\(\frac{{2\pi }}{3}\)
\({135^ \circ }\)	\(\frac{{3\pi }}{4}\)
\({150^ \circ }\)	\(\frac{{5\pi }}{6}\)
\({180^ \circ }\)	\(\pi \)
\({270^ \circ }\)	\(\frac{{3\pi }}{2}\)
\({360^ \circ }\)	\(2\pi \)

Conversion of Fractions and Decimals to Degree-Minute-Second

Sometimes angles are given in fractions or decimals. But it is not practical to take the decimal form and go forward. It is as good as a situation where the length of something is expressed in cm, m and dcm. In such a case, what do we do? We convert all of the different units to the same unit to make it easy to add, subtract, compare etc. In the same way, we want to convert the given angles into their standard form for comparing or solving them.

First, we convert the fractions into decimals if the angle is given in the fractional form. Then further, we will convert them to Degree-Minute-Second. On converting from fraction to decimal, the degree may be with or without decimal part. If there is no decimal part, it means it has only degrees and no minutes and seconds. If a decimal is present, the whole part is in degrees, while the decimal part will convert to minutes and seconds using the relation between degree, minutes and seconds. Let us understand this conversion with the following example.

Examples:

Convert decimal degrees \(156.742\) to degrees-minutes-seconds

The whole part of the decimal number remains in degrees. So \(156.742\) gives you \(156\) degrees.

Now the decimal part has to be converted to minutes by multiplying by \(60\)
\(0.742×60=44.52,\) here the whole number \(44\) is in minutes.

Again the remaining decimal part is multiplied by \(60\) to convert to seconds
\(0.52×60=31.2\), so the whole number \(31\) equals seconds.

Decimal \(156,{742^ \circ }\) is now converted to \(156\) degrees, \(44\) minutes and \(31\) seconds, or \({156^ \circ },44,31”\)

Follow the math rules of rounding when resulting seconds is something like \(31.9.\) You may round up to \(32.\)

Solved Examples on Relationship of Degree with Minute and Second

Q.1. Convert 26.43° into degree-minutes-seconds.
Ans: \({26.43^ \circ } = {26^ \circ } + \left( {0.43 \times {{60}^\prime }} \right)\)
\(= {26^ \circ } + {25.8^\prime }\)
\(= {26^ \circ } + {25^\prime } + \left( {0.8 \times 60} \right)\)
\(= {26^ \circ } + {25^\prime } + {48^{\prime \prime }}\)
\(= {26^ \circ }{25^\prime }{48^{\prime \prime }}\)
Therefore, the angle in degree-minutes-seconds is \({26^ \circ }{25^\prime }{48^{\prime \prime }}\)

Q.2. The measurement of an angle in degrees, minutes and seconds is 84°15′ 45″. Convert to decimal form.
Ans: Given, \({84^ \circ }{15^\prime }{45^{\prime \prime }}\)
\({84^ \circ }{1545^{\prime \prime }} = {84^ \circ } + 15 + {45^{\prime \prime }} \ldots ..(1)\)
\({15^\prime } = {\left( {\frac{{15}}{{60}}} \right)^ \circ } = {0.25^ \circ }\)
\({45^{\prime \prime }} = {\left( {\frac{{45}}{{60 \times 60}}} \right)^ \circ } = {0.0125^ \circ }\)
Substituting these decimal values in \((1)\)
\({84^ \circ }{15^\prime }{45^{\prime \prime }} = {84^ \circ } + {0.25^ \circ } + {0.0125^ \circ }\)
\(= {84.2625^ \circ }\)
Thus, the degree-minutes-second form of an angle is converted to decimal form.

Q.3. Write 34°  in minutes and 21° in seconds.
Ans: From the degree to minute relationship, \({1^ \circ } = {60^\prime }\)
\( \Rightarrow {34^ \circ } = 34 \times {60^\prime } = {2040^\prime }\)
Thus, \(32\) degrees \(=1920\) minutes. 
Now from the degree to second relationship, \({1^ \circ } = 60 \times {60^{\prime \prime }} = 3600”\)
\( \Rightarrow {21^ \circ } = 21 \times 60 \times {60^{\prime \prime }} = {75600^{\prime \prime }}\)
Thus, \(21\) degrees \(= 75600\) seconds.

Q.4. Convert 39600 seconds to degrees.
Ans: From the degree to second relationship, \({1^ \circ } = 60 \times 60” = 3600”\)
By interconversion, \({1^{\prime \prime }} = {\left( {\frac{1}{{3600}}} \right)^ \circ }\)
\( \Rightarrow {39600^{\prime \prime }} = {\left( {\frac{{39600}}{{3600}}} \right)^ \circ } = {11^ \circ }\)
Thus, \(39600\) seconds is equal to \(11\) degrees.

Q.5. Convert 47.72° into degrees-minutes-seconds.
Ans: We first take the decimal part and convert it into degrees-minutes-seconds, then we add the whole part to it.
\({0.72^ \circ } = 0.72 \times {60^\prime }\)
\(=43.2’\) 
\(=43’+(0.2×60″)\)
\(=43’12″\) 
Now, \({47.72^ \circ } = {47^ \circ } + {43^\prime }{12^{\prime \prime }} = {47^ \circ }{43^\prime }{12^{\prime \prime }}\)
Hence, \({47.72^ \circ }\) in degree-minutes-seconds is \({47^ \circ }{43^\prime }12”.\)

Summary

In this article, we studied the relationship of degree with a minute and second, unit of angle measurement. There are three systems of measurement of an angle. They are sexagesimal, centesimal and circular systems. We started from basic definitions: an angle, the definition and the role of a right angle. 

Then, we studied the measurement system of an angle in detail. Also discussed interconversion of sexagesimal and centesimal systems. Further, we discussed the conversion of decimals and fractions to DMS. Examples have been solved for students to make them understand the concept.

FAQs on Relationship of Degree with Minute and Second

Q.1. What is the relation between degree minute and second?
Ans: One degree is divided into \(60\) equal parts known as minutes, and one minute into \(60\) seconds. The use of degrees-minutes-seconds is also called DMS notation.

Q.2. Why are degrees divided into minutes and seconds?
Ans: Degrees are divided further into minutes and seconds to denote the angle between two whole parts of an angle. For example, to write \({23.5^ \circ }\) or \(23\) and a half degrees, we write \({23^ \circ }30\)’ in standard form.

Q.3. How do you prove 1 degree is 60 minutes?
Ans: One degree is split into \(60\) minutes and one minute split into \(60\) seconds. The use of degrees-minutes-seconds is also recognized as DMS notation.

Q.4. What is the relation between longitude and time?
Ans: There exists a close relationship between longitude and time. The Earth makes one complete rotation of \({360^ \circ }\) in \(24\) hours. It passes through \(15\) degrees in an hour or one degree in \(4\) minutes. Thus, there is a difference of \(4\) minutes for one degree of longitude.

Q.5. What is the relation between degree and radian?
Ans: The total number of degrees in one complete rotation is equal to \({360^ \circ }\). In degrees, \(\theta = {360^ \circ }\) The radian measure of this central angle or \(\theta=\frac{s}{r}\) can be found by using the formula for radian measure or \(\theta = \frac{s}{r}\) with \(s=\) length of the intercepted arc.

We hope you find this article on the Relationship of Degree with Minute and Second helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.