SYSTEM OF MEASUREMENT OF ANGLES

Angle Definition

An angle is formed by two rays that with common endpoints, where, ray is the straight line. One ray is called the initial side and other one is called as terminal side.

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There are two systems used for the measurement of angles in trigonometry.

1. Sexagesimal System

2. Circular System

1. Sexagesimal System

Here an angle is measured in degrees, minutes and seconds. In this system, a right angle is divided into equal parts, known as degree, each degree is divided into equal parts known as minutes and each minute is divided into equal parts known as seconds.

Short description

1. 60 seconds or 60’’= minute or

2. 60 minutes or = degree or

3. 90 degrees or 90o= right angle

Example

Express 83.12^\circ in degree, minute and second

\begin{array}{c}
83.12^\circ  = 83^\circ  + \left( {0.12} \right)^\circ \\
 = 83^\circ  + \left( {0.12 \times 60} \right)'\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\therefore \,\,\,\,1^\circ  = 60'} \right)\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 83^\circ  + 7.2'\\
 = 83^\circ  + 7' + \left( {0.2} \right)'\\
 = 83^\circ  + 7' + \left( {0.2 \times 60} \right)''\,\,\,\,\,\,\,\,\,\left( {\therefore \,\,\,1'\, = \,60''} \right)\\
 = 83^\circ  + 7' + 12''
\end{array}

Therefore, 83.12^\circ  =

2. Circular System

Here an angle is measured in radians.

A radian is defined as follows:

A radian corresponds to the angle subtended at the center of a circle by an arc of length which is equal to the radius of the circle.

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Expressed as follows:

{1^{\rm{c}}} = \frac{{{\rm{arc length }}}}{{{\rm{radius }}}}

Since, the arc length of a full circle is the same as the circumference of a circle, which is written as , then, the radian of the full circle (360o)is:

{1^{\rm{c}}} = \frac{{{\rm{arc length }}}}{{{\rm{radius }}}} = \frac{{2\pi r}}{r} = 2\pi

Questions:

Question 1: Express 67.13^\circ in degree, minute and second

Solution

\begin{array}{c}
67.13^\circ  = 67^\circ  + \left( {0.13} \right)^\circ \\
 = 67^\circ  + \left( {0.13 \times 60} \right)'\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {1^\circ  = 60'} \right)\\
 = 67^\circ  + 7.8'\\
 = 67^\circ  + 7' + \left( {0.8} \right)'\\
 = 67^\circ  + 7' + \left( {0.8 \times 60} \right)''\,\,\,\,\,\,\,\,\left( {1' = 60''} \right)\\
 = 67^\circ  + 7' + 48''
\end{array}

Therefore, 67.13^\circ  =

Question 2: Express 50^\circ {\rm{ }}15'{\rm{ }}24'' in radians.

Solution

\begin{array}{c}
50^\circ {\rm{ }}15'{\rm{ }}24'' = 50^\circ  + 15' + 24''\,\,\,\,\,\,\,\,\,\,\,\,\left( {1^\circ  = 60''} \right)\,\,\,\\
 = 50^\circ  + 15' + {\left( {\frac{{24}}{{60}}} \right)^'}\\
 = 50^\circ  + {\left( {15 + \frac{2}{5}} \right)^'}\\
 = 50^\circ  + {\left( {\frac{{77}}{5}} \right)^'}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {1^\circ  = 60'} \right)\\
 = 50^\circ  + {\left( {\frac{{77}}{{300}}} \right)^{\rm{o}}}\\
 = {\left( {\frac{{15077}}{{300}}} \right)^{\rm{o}}}\\
 = \frac{{15077}}{{300}}\, \times \,\frac{\pi }{{180}}\\
 = {\left( {\frac{{15077\pi }}{{54000}}} \right)^c}
\end{array}

Hence, 50^\circ {\rm{ }}15'{\rm{ }}24'' =

Post Author: E-Maths