• Radians or degrees - class-X
Back
0

Radians or degrees - class-X

Description: radians or degrees
Number of Questions: 21
Created by:
Tags: maths mathematics and statistics angle and its measurement sector of a circle and length of arc trigonometric functions and equations angle and their measurement trigonometric functions radians trigonometry trigonometric ratios upto transformations
Start the Quiz
Attempted 0/21 Correct 0 Score 0
‹
›

$\dfrac{\cos(90-A)\sin(90-A)}{\tan (90-A)}$=

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\sin^{2}A$

  2. $\cos^{2}A$

  3. $\sin A$

  4. $1$

Show answer
Correct Option: A
Explanation:
$ =  \dfrac{cos\,(90-A)\,sin\,(90-A)}{tan(90-A)}$
$ = \dfrac{sin\,A \times cos\,A}{\dfrac{cos\,A}{sin\,A}} = \dfrac{sin^{2}A\times cos\,A}{cos\,A}$
$ = sin^{2}A$

The value of $ \displaystyle 36^{\circ} $ in radians is 

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $ \displaystyle \frac{\pi }{2} $

  2. $ \displaystyle \frac{2\pi }{5} $

  3. $ \displaystyle \frac{\pi }{5} $

  4. $ 3\pi $

Show answer
Correct Option: C
Explanation:

$\displaystyle 180^{\circ}      $    =  $\displaystyle \pi       $ radians 


$\displaystyle  \therefore  1^{\circ}=\frac{\pi }{180}    radians     $
$\displaystyle  \Rightarrow 36^{\circ}=\frac{\pi }{180}\times 36 \ radians = \frac{\pi }{5}radians      $

A unit radian is approximately equal to

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $57^{\circ} 17' 43"$

  2. $57^{\circ} 17' 45"$

  3. $57^{\circ} 17' 47"$

  4. $57^{\circ} 17' 49"$

Show answer
Correct Option: B
Explanation:
1 radian x 180 degrees per pi radians = 57.295779513082 degrees.

0.295779513082 degrees x 60 minutes per 1 degree = 17.74677078492 minutes.
0.74677078492 minutes x 60 seconds per 1 minute = 44.8062470952 seconds.

Answer : 57 degrees, 17.75 minutes or

57 degrees, 17 minutes, 45 seconds

The value of $\cot 15^{\circ} \cot 20^{\circ} \cot 70^{\circ} \cot 75^{\circ}$ is equal to

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $-1$

  2. $0$

  3. $1$

  4. $2$

Show answer
Correct Option: C
Explanation:

The value of $\cot 15^{\circ}.\cot 20^{\circ}.\cot 70^{\circ}.\cot 75^{\circ}$ is
$= (\cot 15^{\circ} . \cot 75^{\circ}) (\cot 20^{\circ} \cot 70^{\circ})$

Now $\cot 75^{\circ} = \tan 15^{\circ}$ and $\cot 70^{\circ} = \tan 20^{\circ}$   ....................... $\cot(90-\theta)=\tan\theta$

Therefore, the given expression can be written as 
$ (\cot 15^{\circ} \tan 15^{\circ})(\cot 20^{\circ} \tan 20^{\circ})$
$= 1\times 1 = 1$

Consider the following statements :
1. $1^o$ in radian measure is less than 0.02 radians.
2. 1 radian in degree measure is greater than $45^o$ 
Which of the above statements is/are correct ?

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. 1 only

  2. 2 only

  3. Both 1 and 2

  4. Neither 1 nor 2

Show answer
Correct Option: C
Explanation:

1 degree in radians $=1\times \dfrac { \pi  }{ 180 } =0.0174$  which is less than $0.02$ radians.


1 radians in degree $=1\times \dfrac { 180 }{ \pi  } =57.32$ which is greater than $45^{o}$.
Hence, both are correct.

If $\tan 45^{\circ} = \cot \theta$, then the value of $\theta$, in radians is

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\pi$

  2. $\dfrac{\pi}{9}$

  3. $\dfrac{\pi}{4}$

  4. $\dfrac{\pi}{12}$

Show answer
Correct Option: C
Explanation:

$\tan 45^{\circ} = \cot \theta$
$\tan \theta = \cot \theta$   only for $\theta = \pi/ 4$.
Hence, $\theta = 45^{\circ}$ or $\pi/ 4$.

The value of $cos^{2}30^{0}-cos^{2}60^{0}-cos 60^{0}$ is

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $0$

  2. $\dfrac{1}{2}$

  3. $\dfrac{3}{4}$

  4. $1$

Show answer
Correct Option: A
Explanation:

$cos^{2}30^{0}-cos^{2}60^{0}-cos 60^{0}={ \left( \frac { \sqrt { 3 }  }{ 2 }  \right)  }^{ 2 }-{ \left( \frac { 1 }{ 2 }  \right)  }^{ 2 }-\frac { 1 }{ 2 } =\frac { 3 }{ 4 } -\frac { 1 }{ 4 } -\frac { 1 }{ 2 } =\frac { 3-1-2 }{ 4 } =0$

If $A+B=\dfrac { \pi  }{ 3 } $ and $\cos { A } +\cos { B } =1 $, then which of the following are true: 

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\cos { \left( A-B \right) =\dfrac { 1 }{ 3 } } $

  2. $\cos { \left( A-B \right) =-\dfrac { 1 }{ 3 } } $

  3. $\left| \cos { A } -\cos { B } \right| =\sqrt { 2/3 } $

  4. $\left| \cos { A } -\cos { B } \right| =\cfrac { 1 }{ \sqrt { 3 } } $

Show answer
Correct Option: B,C
Explanation:

Ans. $(b)$, $(c)$

From the given relation, we have 
$2\cos { \cfrac { A+B }{ 2 }  } \cos { \cfrac { A-B }{ 2 } =1 }$
Or $2\cos { {30}^{o} } \cos { \cfrac { A-B }{ 2 }  } =1$
$\therefore\quad \cos { \cfrac { A-B }{ 2 }  } =\cfrac { 1 }{ \sqrt { 3 }  }$
$\therefore\quad \cos { \left( A-B \right)  } =\cos ^{ 2 }{ \cfrac { A-B }{ 2 } -1 } =2.\cfrac { 1 }{ 3 } -1=-\cfrac { 1 }{ 3 } \Rightarrow \left( b \right)$
Again $\left| \cos { A } -\cos { B }  \right| =2\sin { \cfrac { A+B }{ 2 }  } \sin { \cfrac { B-A }{ 2 }  }$ 
$=2\sin { {30}^{o} } \sqrt { 1-\cos ^{ 2 }{ \cfrac { A-B }{ 2 }  }  } =1\sqrt { 1-\cfrac { 1 }{ 3 }  } =\sqrt { \cfrac { 2 }{ 3 }  }$

The angle subtended at the centre of circle of radius $3$ metres by an arc of length $1$ metre is equal to

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $20^\circ $

  2. $60^\circ $

  3. $\dfrac{1}{3}\,radian$

  4. $\,3\,radian$

Show answer
Correct Option: C
Explanation:

We know that 

$l=r\times\theta$

Where $l\rightarrow arc$ $length$
            $r\rightarrow radius$
            $\theta\rightarrow angle$ $subtended$ $by$ $the$ $arc$

Substituting the values of these terms we get,

$\Rightarrow 1=3\times\theta$

$\Rightarrow\theta=\dfrac{1}{3} radian$

The value of $\dfrac{1}{\cos 290^o}+\dfrac{1}{\sqrt{3}\sin 250^o}$ is?

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\dfrac{2\sqrt{3}}{3}$

  2. $\dfrac{4\sqrt{3}}{3}$

  3. $\sqrt{3}$

  4. None

Show answer
Correct Option: B
Explanation:
Here, $\dfrac{1}{ \cos 290^{o}} + \dfrac{1 }{ \sqrt{3} \sin 250^{o}}$ 
$= \dfrac{1}{ \cos (270+20)^{o}} + \dfrac{1}{ \sqrt{3} \sin (270-20)^{o}}$
as we know, $\cos (270+A)= \sin A$
& $\sin (270- B)= - \cos B$
So, $=\dfrac{1}{\sin 20}+ \dfrac{1}{\sqrt{3} (- \cos 20)}$
$=\dfrac{- \sqrt{3} \cos 20+ \sin 20}{- \sqrt{3} \cos 20 \cos 20}$
$=\dfrac{- (\sqrt{3} \cos 20 - \sin 20)}{- \sqrt{3} \sin 20 \cos 20}$
$ =\dfrac{ \sqrt{3} \cos 20- \sin 20}{\sqrt{3} \sin 20 \cos 20}$
(Multiply & Divide in Numerator & denominator by $2$ we get.  )
$=\dfrac{2 \left( \dfrac{\sqrt{3}}{2} \cos 20- \dfrac{1}{2} \sin 20  \right)}{\dfrac{\sqrt{3}}{2} (2 \sin 20 \cos 20)}$
$=\dfrac{2 (\sin 60 \cos 20- \cos 60 \sin 20)}{\dfrac{\sqrt{3}}{2} (\sin 40)}$
$=\dfrac{4}{ \sqrt{3}} \dfrac{\sin (60-20)}{\sin (40)}=\dfrac{4}{\sqrt{3}}= \dfrac{4\sqrt{3}}{3} $
So, value is $4 \sqrt{3}/3$

Find the degree measure corresponding to $\left(\dfrac{1}{6}\right)^C$.

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $9.549^\circ$

  2. $9^032$ $43.6"$

  3. $10^0$

  4. None

Show answer
Correct Option: A
Explanation:

$\pi$ rad $=180^o$

$\therefore 1\ rad=\dfrac{180}{\pi}=57.296^o$
$\therefore (1/6)^c=1/6\times \dfrac{180}{\pi}=9.549^o$

If $\cos x=\sqrt{1-\sin2x},0\le x\le \pi$, then possible  value of $x$ is 

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\pi$

  2. $0$

  3. $\tan^{-1}2$

  4. $3\pi$

Show answer
Correct Option: B,C
Explanation:

$\cos { x } =\sqrt { 1-\sin { 2x }  } $; $x\in (0,\pi)$

$=\sqrt { 1-2\sin { x } .\cos { x }  } =\sqrt { \sin ^{ 2 }{ x } -2\sin { x } \cos { x } +\cos ^{ 2 }{ x }  } \left[ \because 1=\sin ^{ 2 }{ x } +\cos ^{ 2 }{ x } ,\forall x\in R \right] $
$=\sqrt { { \left( \sin { x } -\cos { x }  \right)  }^{ 2 } } \left[ \because \sqrt { { x }^{ 2 } } =\left| x \right|  \right] $
$\cos { x } =\left| \sin { x } -\cos { x }  \right| $
case I
$\sin { x } \ge \cos { x } ,x\in \left[ 0,\pi  \right] \Rightarrow \left| \sin { x } -\cos { x }  \right| =\sin { x } -\cos { x } $
$\therefore \log { x } =\sin { x } -\cos { x } $
$\therefore \cos { x } =\sin { x } -\cos { x } \Rightarrow 2\cos { x } =\sin { x } \Leftrightarrow \tan { x } =2\Rightarrow x=\tan ^{ -1 }{ 2 } \left[ \because x\in \left[ 0,\pi  \right]  \right] $
case II
$\sin { x } <\cos { x } ;x\in \left[ 0,\pi  \right] $
$\Rightarrow \left| \sin { x } -\cos { x }  \right| =\cos { x } -\sin { x } $
$\therefore \cos { x } =\cos { x } -\sin { x } \Rightarrow \sin { x } =0\Rightarrow x=0,\pi $
but $x=\pi$ is rejected as $\cos (\pi)=-1$
$\therefore$ only $x=0$
Finally $x=\tan ^{ -1 }{ 2 } ,0$

The area of a sector of a circle of radius $7\ cm$ and central angle $120^{o}$ is 

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $152\ cm^{2}$

  2. $\dfrac{154}{3}\ cm^{2}$

  3. $\dfrac{128}{3}\ cm^{2}$

  4. $128\ cm^{2}$

Show answer
Correct Option: B
Explanation:
Area$=\cfrac { 120 }{ 360 } \times \pi { r }^{ 2 }$
$=\cfrac { \pi  }{ 3 } \times 7\times 7=49\times \cfrac { \pi  }{ 3 } $
$=49\times \cfrac { 22 }{ 7\times 3 } =\cfrac { 154 }{ 3 }cm^2$

$\displaystyle \frac{\pi ^{c}}{5}$ in sexagesimal measure is _____

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\displaystyle 18^{\circ}$

  2. $\displaystyle 36^{\circ}$

  3. $\displaystyle 54^{\circ}$

  4. $\displaystyle 72^{\circ}$

Show answer
Correct Option: B
Explanation:

In $\text{Sexagesimal System}$, an angle is measured in degrees, minutes and seconds.
$ \pi = {180}^{0} $

So, $ \dfrac {\pi}{5} = \dfrac {{180}^{0}}{5} = {36}^{0}  $

The value of $\displaystyle 144^{\circ}$ in circular measure is ___ 

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\displaystyle \frac{3\pi ^{c}}{4}$

  2. $\displaystyle \frac{2\pi ^{c}}{3}$

  3. $\displaystyle \frac{4\pi ^{c}}{5}$

  4. $\displaystyle \frac{5\pi ^{c}}{6}$

Show answer
Correct Option: C
Explanation:

$ {144}^{0} = {144}^{0} \times \dfrac {{\pi}^{c}}{{180}^{0}} = \dfrac {4{\pi}^{c}}{5} $

Find the angle measure of $4$ radians.

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $114.591^{\circ}$

  2. $141.372^{\circ}$

  3. $229.183^{\circ}$

  4. $282.743^{\circ}$

  5. $458.366^{\circ}$

Show answer
Correct Option: C
Explanation:

We know, $\pi$ radians is equal to $180$ degrees
$\therefore 4$ radians is equal to $4 \times \dfrac {180 }{\pi} = 180 \times 1.2733 = 229.183$ degrees

1 radian =

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $58^0.17$

  2. $65^0.17$

  3. $57^0.30$

  4. $66^0.71$

Show answer
Correct Option: C
Explanation:
We know that, $\pi$ radian$=180^o$

so $1$ radian$=\left(\dfrac{180}{\pi}\right)^o$

$=\left(\dfrac{180}{22/7}\right)^o$

$=(57.30)^0$

If $\displaystyle\cot  \theta+\left ( \frac{1}{\sqrt{3}} \right )\sin \theta =\frac{2}{\sqrt{3}} $ then find $\displaystyle \theta $ in circular measure

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\displaystyle \frac{\pi ^{2}}{2} $

  2. $\displaystyle \frac{\pi ^{2}}{3} $

  3. $\displaystyle \frac{\pi ^{2}}{4} $

  4. $\displaystyle \frac{\pi ^{2}}{6} $

Show answer
Correct Option: D

The degree measure of 1 radian (taking $\pi =\dfrac { 22 }{ 7 }$ ) is

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $55^o{ 61 }^{ ' }{ 22 }^{ " }$ (approx.)

  2. $57^o{ 16 }^{ ' }{ 22 }^{ " }$ (approx.)

  3. $57^o{ 22 }^{ ' }{ 16 }^{ " }$ (approx.)

  4. $57^o{ 22 }^{ ' }{ 22 }^{ " }$ (approx.)

Show answer
Correct Option: B
Explanation:

$\pi\ radians = 180^{\circ}$

$1\ radian=\frac { 180 }{ \pi  } = \frac { 180 }{ \frac { 22 }{ 7 }  } $
$1\ radian=57.272727$
The integer part constitutes the degree part. The mantissa is converted to minutes by multiplying with ${60}'$
Minutes = $0.272727*{60}' = {16.3636}'$
The integer part constitutes the minutes. The mantissa is converted to seconds by multiplying with ${60}''$
Seconds = $0.3636*{60}''\approx {22}''$
Hence, the degree measure of 1 radian is $57^{\circ}{16}'{22}''$

Convert $40^\circ \,20'$ into radian measure.

mathematics and statistics angle and its measurement degree measure of angle measure of angle radians or degrees

  1. $\dfrac {121}{540}\pi $ radians

  2. $\dfrac {121}{570}\pi $ radians

  3. $\dfrac {120}{513}\pi $ radians

  4. None

Show answer
Correct Option: A
Explanation:
Given: ${40^0}$${{{20}^{'}}}$

${40^0} + \dfrac{{{{20}^0}}}{{{{60}^0}}}  $

$=40 + \dfrac{1^o}{3} = \dfrac{{{{121}^0}}}{3}$

$radian = \dfrac{\pi }{{180^o}} \times \dfrac{{121^o}}{3}$

$\boxed{ = \dfrac{{121}}{{540}}\pi \;radians}$

Find the radian measure corresponding to the degree $-47^{o}30'$

mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

  1. $\dfrac {-19\ \pi}{72}rad$

  2. $\dfrac {19\ \pi}{72}rad$

  3. $\dfrac {13\ \pi}{72}rad$

  4. $None\ of\ these$

Show answer
Correct Option: A
Explanation:
$-47^{o} 30'$
$\Rightarrow - (47+ \dfrac{30}{60}) (\because 1^{o} =60')$
$\Rightarrow  -\left( 47+ \dfrac{1}{2} \right)$
$-\left( \dfrac{95}{2} \right)$
Radian measure $\Rightarrow \dfrac{\pi}{180} \times \dfrac{-95}{2}$
$\Rightarrow \pi x - \dfrac{19}{72} \Rightarrow - \dfrac{19 \pi}{72}$ radian
- Hide questions
ADVERTISEMENT

Find more quizzes from top tags

general knowledge
3615 Quizzes
119858 Questions
 technology
307 Quizzes
36705 Questions
 sports
963 Quizzes
19148 Questions
 history
1187 Quizzes
13475 Questions
 physics
598 Quizzes
13441 Questions
 biology
1160 Quizzes
12174 Questions
 science & technology
551 Quizzes
11015 Questions
 chemistry
490 Quizzes
9892 Questions
 maths
518 Quizzes
9324 Questions
 softskills
0 Quizzes
7131 Questions