1. Find the radian measures corresponding to the following degree measures:
(i) 25°
$\displaystyle \small 25^{\circ}=\left ( \frac{\pi }{180} \right )*25=\frac{5\pi }{36}$ radian
(ii) – 47°30′
$\displaystyle \small -47^{\circ}{30}'=-\left [ 47^{\circ}+\left ( \frac{30}{60} \right )^{\circ} \right ]$
$\displaystyle \small =-\left [ 47^{\circ}+\left ( \frac{1}{2} \right )^{\circ} \right ]$
$\displaystyle \small =-\left ( \frac{95}{2} \right )^{\circ}$
$\displaystyle \small =-\left ( \frac{95}{2} \right )^{\circ}$ =$\displaystyle \small -\left ( \frac{\pi }{180}*\frac{95}{2} \right )=-\left ( \frac{19\pi }{72} \right )$ radian
(iii) 240°
$\displaystyle \small 240^{\circ}=\frac{\pi }{180}*240=\frac{4\pi }{3}$ radian
(iv) 520°
$\displaystyle \small 520^{\circ}=\frac{\pi }{180}*520=\frac{26\pi }{9}$ radian
2.Find the degree measures corresponding to the following radian measures (Use π=22/7).
(i) $\displaystyle \small \frac{11}{16}$
$\displaystyle \small \frac{11}{16} radian = \frac{180}{\pi }*\frac{11}{16}$
$\displaystyle \small = \frac{180}{22 }*7*\frac{11}{16}=\frac{315}{8}$
$\displaystyle \small =39^{\circ}+\left ( \frac{3}{8} \right )^{\circ}$
$\displaystyle \small =39^{\circ}+{\left ( \frac{3}{8}*60 \right )}'$
$\displaystyle \small =39^{\circ}+{22}'+{\left ( \frac{4}{8} \right )}'$
$\displaystyle \small =39^{\circ}+{22}'+{\left ( \frac{4}{8}*60 \right )}''$
$=39^{\circ}{22}'{30}''$
(ii) -4
$\displaystyle \small =-\left ( \frac{180}{\pi } *4\right )=-\left ( \frac{180}{22}*7*4 \right )$
$\displaystyle \small =-\left ( \frac{2520}{11} \right )^{\circ}=-\left [ 2520^{\circ}+\left ( \frac{1}{11} \right )^{\circ} \right ]$
$\displaystyle \small =-\left [ 2520^{\circ}+{\left ( \frac{1}{11}*60 \right )}' \right ]$
$\displaystyle \small =-\left [ 2520^{\circ}+{5}'+\left ({\frac{5}{11}} \right )' \right ]$
$\displaystyle \small =-\left [ 2520^{\circ}+{5}'+{\left ({\frac{5}{11}*60} \right )}'' \right ]$
$=2520^{\circ}{5}'{27}''$
(iii) $\displaystyle \small \frac{5\pi }{3}$
$\displaystyle \small =\left ( \frac{180}{\pi }*\frac{5\pi }{3} \right )^{\circ}=300^{\circ}$
(iv) $\displaystyle \small \frac{7\pi }{6}$
$\displaystyle \small =\left ( \frac{180}{\pi }*\frac{7\pi }{6} \right )^{\circ}=210^{\circ}$
3. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
Number of revolutions made in 1min = 360
∴ Number of revolutions made in 1 sec = 360/60 = 6
One complete revolution = 2π radian
6 complete revolution = 6*2π = 12π radian
4. Find the degree measure of the angle subtended at the center of a circle of radius 100cm by an arc of length 22 cm (Use π=22/7).
Given that, r=100cm, s= 22cm
We know that , $\displaystyle \small \theta =\frac{s}{r}$
$\displaystyle \small =\frac{22}{100}$ radian
$\displaystyle \small =\left (\frac{180}{\pi }*\frac{22}{100} \right )^{\circ}=\left (\frac{180*7*22}{22*100} \right )^{\circ}$
$\displaystyle \small =\left ( \frac{126}{10} \right )^{\circ}=12^{\circ}+\left ( \frac{3}{5} \right )^{\circ}$
$\displaystyle \small =12^{\circ}+{\left ( \frac{3}{5} *60\right )}'=12^{\circ}+{36}'$
$=12^{\circ}{36}'$
5. In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
We have radius, r = 40/2 = 20cm
length of chord = 20cm [△OAB is an equilateral triangle]
∴ θ = 60°= $\displaystyle \small 60*\frac{\pi }{180}=\frac{\pi }{3}$ radian
We know that, $\displaystyle \small \theta =\frac{arc AB}{r}$
arc AB = rθ = 20*$\displaystyle \small \frac{\pi }{3}=\frac{20\pi }{3}$ cm
6. If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.
Let the radii of the two circles be r1 and r2 respectively. Also, length of arc in each case be l.
We know that, $\displaystyle \small \theta =\frac{l}{r}$
⇒ $\displaystyle \small r=\frac{l}{\theta }$
For first circle,
$\displaystyle \small \theta =60^{\circ}=60*\frac{\pi }{180}=\frac{\pi }{3}$ radian
$\displaystyle \small r_{1}=\frac{l}{\theta }=\frac{3l}{\pi }$
For second circle,
$\displaystyle \small \theta =75^{\circ}=75*\frac{\pi }{180}=\frac{5\pi }{12}$ radian
$\displaystyle \small r_{2}=\frac{l}{\theta }=\frac{12l}{5\pi }$
The ratio of radii is,
$\displaystyle \small \frac{r_{1}}{r_{2}}=\frac{3l/\pi }{12l/5\pi }=\frac{5}{4}$
7. Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length
(i) 10 cm (ii) 15 cm (iii) 21 cm
It is given that r = 75cm
We know that, $\displaystyle \small \theta =\frac{l}{r}$
(i) l = 10cm
$\displaystyle \small \theta =\frac{l}{r}=\frac{10}{75}=\frac{2}{15}$ radian
(ii) l = 15cm
$\displaystyle \small \theta =\frac{l}{r}=\frac{15}{75}=\frac{1}{5}$ radian
(iii) l = 21cm
$\displaystyle \small \theta =\frac{l}{r}=\frac{21}{75}=\frac{7}{25}$ radian
(i) 25°
$\displaystyle \small 25^{\circ}=\left ( \frac{\pi }{180} \right )*25=\frac{5\pi }{36}$ radian
(ii) – 47°30′
$\displaystyle \small -47^{\circ}{30}'=-\left [ 47^{\circ}+\left ( \frac{30}{60} \right )^{\circ} \right ]$
$\displaystyle \small =-\left [ 47^{\circ}+\left ( \frac{1}{2} \right )^{\circ} \right ]$
$\displaystyle \small =-\left ( \frac{95}{2} \right )^{\circ}$
$\displaystyle \small =-\left ( \frac{95}{2} \right )^{\circ}$ =$\displaystyle \small -\left ( \frac{\pi }{180}*\frac{95}{2} \right )=-\left ( \frac{19\pi }{72} \right )$ radian
(iii) 240°
$\displaystyle \small 240^{\circ}=\frac{\pi }{180}*240=\frac{4\pi }{3}$ radian
(iv) 520°
$\displaystyle \small 520^{\circ}=\frac{\pi }{180}*520=\frac{26\pi }{9}$ radian
2.Find the degree measures corresponding to the following radian measures (Use π=22/7).
(i) $\displaystyle \small \frac{11}{16}$
$\displaystyle \small \frac{11}{16} radian = \frac{180}{\pi }*\frac{11}{16}$
$\displaystyle \small = \frac{180}{22 }*7*\frac{11}{16}=\frac{315}{8}$
$\displaystyle \small =39^{\circ}+\left ( \frac{3}{8} \right )^{\circ}$
$\displaystyle \small =39^{\circ}+{\left ( \frac{3}{8}*60 \right )}'$
$\displaystyle \small =39^{\circ}+{22}'+{\left ( \frac{4}{8} \right )}'$
$\displaystyle \small =39^{\circ}+{22}'+{\left ( \frac{4}{8}*60 \right )}''$
$=39^{\circ}{22}'{30}''$
(ii) -4
$\displaystyle \small =-\left ( \frac{180}{\pi } *4\right )=-\left ( \frac{180}{22}*7*4 \right )$
$\displaystyle \small =-\left ( \frac{2520}{11} \right )^{\circ}=-\left [ 2520^{\circ}+\left ( \frac{1}{11} \right )^{\circ} \right ]$
$\displaystyle \small =-\left [ 2520^{\circ}+{\left ( \frac{1}{11}*60 \right )}' \right ]$
$\displaystyle \small =-\left [ 2520^{\circ}+{5}'+\left ({\frac{5}{11}} \right )' \right ]$
$\displaystyle \small =-\left [ 2520^{\circ}+{5}'+{\left ({\frac{5}{11}*60} \right )}'' \right ]$
$=2520^{\circ}{5}'{27}''$
(iii) $\displaystyle \small \frac{5\pi }{3}$
$\displaystyle \small =\left ( \frac{180}{\pi }*\frac{5\pi }{3} \right )^{\circ}=300^{\circ}$
(iv) $\displaystyle \small \frac{7\pi }{6}$
$\displaystyle \small =\left ( \frac{180}{\pi }*\frac{7\pi }{6} \right )^{\circ}=210^{\circ}$
3. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
Number of revolutions made in 1min = 360
∴ Number of revolutions made in 1 sec = 360/60 = 6
One complete revolution = 2π radian
6 complete revolution = 6*2π = 12π radian
4. Find the degree measure of the angle subtended at the center of a circle of radius 100cm by an arc of length 22 cm (Use π=22/7).
Given that, r=100cm, s= 22cm
We know that , $\displaystyle \small \theta =\frac{s}{r}$
$\displaystyle \small =\frac{22}{100}$ radian
$\displaystyle \small =\left (\frac{180}{\pi }*\frac{22}{100} \right )^{\circ}=\left (\frac{180*7*22}{22*100} \right )^{\circ}$
$\displaystyle \small =\left ( \frac{126}{10} \right )^{\circ}=12^{\circ}+\left ( \frac{3}{5} \right )^{\circ}$
$\displaystyle \small =12^{\circ}+{\left ( \frac{3}{5} *60\right )}'=12^{\circ}+{36}'$
$=12^{\circ}{36}'$
5. In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
length of chord = 20cm [△OAB is an equilateral triangle]
∴ θ = 60°= $\displaystyle \small 60*\frac{\pi }{180}=\frac{\pi }{3}$ radian
We know that, $\displaystyle \small \theta =\frac{arc AB}{r}$
arc AB = rθ = 20*$\displaystyle \small \frac{\pi }{3}=\frac{20\pi }{3}$ cm
6. If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.
We know that, $\displaystyle \small \theta =\frac{l}{r}$
⇒ $\displaystyle \small r=\frac{l}{\theta }$
For first circle,
$\displaystyle \small \theta =60^{\circ}=60*\frac{\pi }{180}=\frac{\pi }{3}$ radian
$\displaystyle \small r_{1}=\frac{l}{\theta }=\frac{3l}{\pi }$
For second circle,
$\displaystyle \small \theta =75^{\circ}=75*\frac{\pi }{180}=\frac{5\pi }{12}$ radian
$\displaystyle \small r_{2}=\frac{l}{\theta }=\frac{12l}{5\pi }$
The ratio of radii is,
$\displaystyle \small \frac{r_{1}}{r_{2}}=\frac{3l/\pi }{12l/5\pi }=\frac{5}{4}$
7. Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length
(i) 10 cm (ii) 15 cm (iii) 21 cm
We know that, $\displaystyle \small \theta =\frac{l}{r}$
(i) l = 10cm
$\displaystyle \small \theta =\frac{l}{r}=\frac{10}{75}=\frac{2}{15}$ radian
(ii) l = 15cm
$\displaystyle \small \theta =\frac{l}{r}=\frac{15}{75}=\frac{1}{5}$ radian
(iii) l = 21cm
$\displaystyle \small \theta =\frac{l}{r}=\frac{21}{75}=\frac{7}{25}$ radian
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