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This is one of our many quizzes on Radians. Click on this to access more such competitive quizzes on Trigonometry.

Radians Quiz 2

mathematics

 Measure the value of radius from ${{30}^{\circ }},\text{ 21 }\!\!'\!\!\text{  25'' }$.

 Firstly, convert to degree form and further convert to radian form.

${{30}^{\circ }},\text{ 21 }\!\!'\!\!\text{  25''} \\  \text{3}{{\text{0}}^{\circ }}+21'+25'' \\  \left( 30+\dfrac{21}{60}+\dfrac{25}{3600} \right)\dfrac{\pi }{180} \\  \Rightarrow \left( \dfrac{21600+252+5}{720} \right)\times \dfrac{\pi }{180} \\  =\dfrac{21857\pi }{129600}. \\ $

 A person is run around a circular park with speed $10m{{s}^{-1}}$ for $3$ minute, what would be the central be angle made at the centre of the park, if the radius of the park is $50m$.

 Calculate the value of distance by applying formula $D=S\times T$ and then apply formula $\ell =r\theta $.

$ \text{Distance = speed  }\!\!\times\!\!\text{  time} \\  \therefore \text{=10 }\!\!\times\!\!\text{ 3 }\!\!\times\!\!\text{ 60} \\  \Rightarrow \text{1800m} \\  \text{Arc length (}\ell \text{) = 1800m} \\  \text{r=50m} \\  \ell \text{=r}\times \theta  \\  \text{1800=50}\times \theta  \\  \theta =36\text{ radian} \\ $

 If the radius of the circle is $20cm$ and length of arc $40cm$. Find angle of centre in radius form.

 Apply length of arc formula, $L = \dfrac{\theta }{{360}} \times 2\pi r$

$ \text{Given radius of circle 20cm} \\  \text{length of arc = 40cm} \\  \dfrac{\theta }{360}\times 2\pi r=40cm \\  \theta =\dfrac{40\times 360}{2\times 20} \\  \theta =\dfrac{360}{\pi } \\  \text{in radian form =}\dfrac{360}{\pi }\times \dfrac{\pi }{180}=2 \\ $

 Evaluate the value of angle in radian subtended by the hour of clock if the time changes from $7:45$ am to $7:59$ am.

 Calculate the angle formed by hour hand in 1 minute and then further solve.

$ \text{time = 7:59-7:45} \\  \Rightarrow \text{14min} \\  \text{Angle subtended by hour hand 12 hours = 36}{{\text{0}}^{\circ }} \\  1\text{ angle subtended by hand 1 hours = }\dfrac{{{360}^{\circ }}}{12}={{30}^{{}^\circ }} \\  \text{Angle in 1 min}\text{. =}\dfrac{{{30}^{{}^\circ }}}{{{60}^{{}^\circ }}}=\dfrac{1}{2} \\  \text{Angle in 14m = 14 }\!\!\times\!\!\text{ }\dfrac{1}{2}=7 \\  \therefore \text{Radian = 7 }\!\!\times\!\!\text{ }\dfrac{\pi }{{{180}^{{}^\circ }}}=\dfrac{7\pi }{{{180}^{{}^\circ }}} \\ $

 Find the area of shaded region if chord is $\dfrac{1}{2}$ time of radius and centre angle ${{120}^{\circ }}$ if the area of triangle is $2$ unit square. $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/45fb3d4c-c61c-41e7-9014-ed867f0a6ed81383861608539377591.png"/>

 First calculate the radius of the circle and then use the area subtended by an arc.

$  \text{Let r be the radius of circle (AO)} \\  \therefore \text{chord = }\dfrac{1}{2}\times r=\dfrac{r}{2} \\  \theta ={{120}^{\circ }} \\  In\text{ }\Delta \text{AOM} \\  \text{AM=MB=}\dfrac{r}{4} \\  SinA=\dfrac{OM}{AO} \\  \therefore Sin30=\dfrac{OM}{r} \\  \dfrac{r}{2}=OM \\  \text{area of }\Delta \text{=}\dfrac{1}{2}\times B\times h \\  \therefore 2=\dfrac{1}{2}\times \dfrac{r}{2}\times \dfrac{r}{2} \\  \Rightarrow r=4 \\  \text{area of shaded region} \\  \text{=}\dfrac{\theta }{360}\pi {{r}^{2}}-area\text{ }\left( \Delta AOB \right) \\  =\dfrac{120}{360}\times \dfrac{22}{7}\times 16-2 \\ \dfrac{120}{360}\times \dfrac{22}{7}\times 16-2 \\  =\dfrac{310}{21}\text{unit square} \\ =\frac{352}{21}-2 \\ =\dfrac{310}{21}\text{unit square} \\ $