Measurement of Angles

Measurement of Angles


Geometry measures angle in terms of Right Angle. Now, the right angle is a large unit. We needed to divide it into smaller units viz. degrees. Thus arose different systems of measurement of angles. Skip to Examples

Sexagesimal System Centesimal System
1 Right Angle \( = 90^0 \) 1 Right Angle \( = 100^g \)
\(1^0 = 60^\prime\) \(1^g = 100^\prime\)
\(1^\prime = 60^{\prime\prime}\) \(1^\prime = 100^{\prime\prime}\)

Ironically, although the Centesimal system is more in sync with the decimal system it is the Sexagesimal system that is in universal practice. To say the least, it is odd on account of 60 and 90 being the multipliers. Anyway, it is no more inconvenient due to it being in use for long.
To convert:\[1^0 = \frac{10^g}{9}\] \[1^g = \frac{9^0}{10}\] In general: \[x^0 = \left( x + \frac{x}{9} \right)^g\] \[x^g = \left( x – \frac{x}{10} \right)^0\]

Circular Measure

Radian is the unit of circular measurement in higher branches of mathematics.
\[1^c = 57^017^{{}^{\prime}}44.8^{{}^{\prime}{}^{\prime}}\]\[\pi^c = 2\text{ Right Angles } = 180^0 = 200^g\] In print, \(\pi^c\) is written simply as \(\pi\), without symbol. However \(\pi^c\) and \(\pi\) are two different entities and are not to be confused. \(\pi\) is just a number, a constant ratio of \(\frac{circumference}{diameter}\).

\(\pi\) is not a whole number nor can it be expressed as vulgar fraction nor deducible to decimal fraction, terminating or non-terminating. The popular fraction \(\frac{22}{7}\) gives the value of \(\pi\) correctly for the first two decimal places. A better fraction \(\frac{355}{113}\) is correct up to 6 decimal places. Let’s sum up the values: \[\pi = 3.14159265..\]\[\frac{1}{\pi} = 0.318309..\]

Examples

  1. How many degrees, minutes and seconds are respectively passed over in \(11\frac{1}{9}\) minutes by the hour and minute hands of a wrist-watch and a wall clock?
  2. The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other; express both the angles in degrees.
  3. The radius of a carriage wheel is 50 cm, and in \(\frac{1}{9}\)th of a second, it turns through \(80^0\) about its centre that is fixed. How many km does a point on the rim of the wheel travel in one hour?
  4. The circular measure of two angles of a triangle are \(\frac{1}{2}\) and \(\frac{1}{3}\) respectively; what is the number of degrees in the third angle?
  5. The angles of a triangle are in A.P and the number of degrees in the least is to the number of radians in the greatest as 60 to \(\pi\); find the angles in degrees.
  6. What is the ratio of the radii of two circles at the centre of which two arcs of the same length subtend angles of \(60{°}\) and \(75{°}\)?

Answers

  1. \( 5{°}33{‘}20{”}\); \(66{°}40{‘}\)
  2. \(47\frac{7}{19}^0\), \(42\frac{12}{19}^0\)
  3. 23 km nearly
  4. \( 132{°}15{‘}12.6{”}\)
  5. \( 30{°}, 60{°}, 90{°}\)
  6. \(5 : 4\)

Study: Complex Numbers – Beyond Real Numbers

Hints

#2. Let \(\angle 1 = x^0\) then, \(\angle 2 = x^g = (90 – x)^0\)

Converting grade into degrees,

\(\frac{9x}{10}^0 = (90 – x)^0\)

#3. Step 1: \(2\pi r = 100\pi\) cm

Step 2: \( 80{°}\) in \(\frac{1}{9}\) sec
\( 360{°}\) in \(\frac{1}{2}\) sec

Step 3: \(\frac{1}{2}\) sec : \(100\pi\) cm
1 hr : \(\frac{100 \times 2 \times 60 \times 60}{100 \times 1~000} \times 3.14159 = 23\) km nearly

#4. We know \(\pi^c = 180^0\)

\(\Rightarrow \frac{1}{2}^c = \frac{90}{\pi}^0 \Rightarrow \frac{1}{3}^c = \frac{60}{\pi}^0\)

Sum of the two \(\angle s = \frac{150}{\pi}^0 = 47.7465^0\)

So, third \(\angle = 132.2535{°} = 132{°}15{‘}12.6{”}\)

#5. Let the angles be \((x – y){°}\), \(x{°}\)and \((x + y){°}\)

then, \(x = 60\), and angles will be \((60 – y){°}\), \(60{°}\)and \((60 + y){°}\)

Converting: \((60 + y){°} = \frac{\pi}{180}(60 + y)^c\)

As per conditions,

\((60 – y){°} : \frac{\pi (60 + y)^c}{180} : : 60 : \pi\)

\(\Rightarrow y = 30\)

#6. \(\frac{x}{r_1} = 60^0 = \frac{\pi}{3}^c\) and \(\frac{x}{r_2} = 75^0 = \frac{5\pi}{12}^c\)

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