Tag: degree measure of angle

Quizzes (0) Questions (22)

Questions Related to degree 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}}}  $

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

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

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