Tag: measure of angle

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Questions Related to measure of angle

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

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

  1. $\pi$
  2. $0$
  3. $\tan^{-1}2$
  4. $3\pi$
Reveal answer Fill a bubble to check yourself
B,C Correct answer
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 $x=\tan ^{ -1 }{ 2 } ,0$

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

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

  1. $152\ cm^{2}$
  2. $\dfrac{154}{3}\ cm^{2}$
  3. $\dfrac{128}{3}\ cm^{2}$
  4. $128\ cm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Area$=\cfrac { 120 }{ 360 } \times \pi { r }^{ 2 }$
$=\cfrac { \pi  }{ 3 } \times 7\times 7=49\times \cfrac { \pi  }{ 3 } $
$=49\times \cfrac { 22 }{ 7\times 3 } =\cfrac { 154 }{ 3 }cm^2$
Multiple choice mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

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

  1. $\displaystyle 18^{\circ}$
  2. $\displaystyle 36^{\circ}$
  3. $\displaystyle 54^{\circ}$
  4. $\displaystyle 72^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
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}  $

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

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

  1. $\displaystyle \frac{3\pi ^{c}}{4}$
  2. $\displaystyle \frac{2\pi ^{c}}{3}$
  3. $\displaystyle \frac{4\pi ^{c}}{5}$
  4. $\displaystyle \frac{5\pi ^{c}}{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

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

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

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

  1. $55^o{ 61 }^{ ' }{ 22 }^{ " }$ (approx.)
  2. $57^o{ 16 }^{ ' }{ 22 }^{ " }$ (approx.)
  3. $57^o{ 22 }^{ ' }{ 16 }^{ " }$ (approx.)
  4. $57^o{ 22 }^{ ' }{ 22 }^{ " }$ (approx.)
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $57^{\circ}{16}'{22}''$

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

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

  1. $\dfrac {121}{540}\pi $ radians
  2. $\dfrac {121}{570}\pi $ radians
  3. $\dfrac {120}{513}\pi $ radians

  4. None
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given: ${40^0}$${{{20}^{'}}}$

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

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

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

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