Mathematics · Quantitative Aptitude

Trigonometric Identities and Values

74 Questions

Trigonometric identities and values questions require evaluating complex angles using standard formulas. These are critical for quantitative aptitude sections in SSC and various state exams. Success depends on memorizing standard values and applying transformation rules.

angle transformationstrigonometric ratiosstandard angle valuessine cosine products

Trigonometric Identities and Values Questions

Multiple choice telescopic summation for infinte series binomial theorem, sequence and series maths

${\sin ^2}{{\text{2}}^{\text{o}}} + {\sin ^2}{{\text{4}}^{\text{o}}} + \;{\sin ^2}{{\text{6}}^{\text{o}}} + \;.... + \;{\sin ^2}{\text{9}}{{\text{0}}^{\text{o}}}$ is equal to

  1. $22$
  2. $23$
  3. $44$
  4. $45$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Now,

${\sin ^2}{{\text{2}}^{\text{o}}} + {\sin ^2}{{\text{4}}^{\text{o}}} + \;{\sin ^2}{{\text{6}}^{\text{o}}} + \;.... + \;{\sin ^2}{\text{9}}{{\text{0}}^{\text{o}}}$
$=({\sin ^2}{{\text{2}}^{\text{o}}} + {\sin ^2}{{\text{4}}^{\text{o}}} + \;{\sin ^2}{{\text{6}}^{\text{o}}} + \;...+\sin^2 44^o)+(\sin^2 46^o+... +\sin^2 98^o)+ \;{\sin ^2}{\text{9}}{{\text{0}}^{\text{o}}}$
$=({\sin ^2}{{\text{2}}^{\text{o}}} + {\sin ^2}{{\text{4}}^{\text{o}}} + \;{\sin ^2}{{\text{6}}^{\text{o}}} + \;...+\sin^2 44^o)+(\cos^2 44^o+... +\cos^2 2^o)+ 1$ [ Since $\sin^2 x^o=\cos^2 (90^o-x^o)$]
$=(\sin^2 2^o+\cos^2 2^o)+(\sin^2 4^o+\cos^2 4^o)+......+(\sin^2 44^o+\cos^2 44^o)+1$
$=1+1+.....+1(22\text{th})+1$
$=22+1$
$=23$.

Multiple choice telescopic summation for infinte series binomial theorem, sequence and series maths

The value of ${ 1 }^{ 2 }.{ _{  }^{ 20 }{ C } } _{ 1 }+{ 2 }^{ 2 }.{ _{  }^{ 20 }{ C } } _{ 2 }+{ 3 }^{ 2 }.{ _{  }^{ 20 }{ C } } _{ 3 }+.....{ (20) }^{ 2 }.{ _{  }^{ 20 }{ C } } _{ 20 }$ is

  1. $210\times { 2 }^{ 17 }$
  2. $420\times { 2 }^{ 17 }$
  3. $420\times { 2 }^{ 87 }$
  4. $210\times { 2 }^{ 87 }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$S={ 1 }^{ 2 }.{ _{  }^{ 20 }{ C } } _{ 1 }+{ 2 }^{ 2 }.{ _{  }^{ 20 }{ C } } _{ 2 }+{ 3 }^{ 2 }.{ _{  }^{ 20 }{ C } } _{ 3 }+.....{ (20) }^{ 2 }.{ _{  }^{ 20 }{ C } } _{ 20 }=\sum _{ r=1 }^{ 20 }{ { r }^{ 2 } } .{ _{  }^{ 20 }{ C } } _{ r }$
$=\sum _{ r=1 }^{ 20 }{ { r }^{  } } (r.{ _{  }^{ 20 }{ C } } _{ r })\=20\sum _{ r=1 }^{ 20 }{ { r }^{ 19 } } .{ _{  }^{ 19 }{ C } } _{ r-1 }\=20\sum _{ r=1 }^{ 20 }{ (r-1+1) } .{ _{  }^{ 19 }{ C } } _{ r-1 }\=20\sum _{ r=1 }^{ 20 }{ { (r-1) }^{  } } .{ _{  }^{ 19 }{ C } } _{ r-1 }+20\sum _{ r=1 }^{ 20 }{ { r }^{ 2 } } .{ _{  }^{ 19 }{ C } } _{ r-1 }\=20\times 19\sum _{ r=2 }^{ 20 }{ { _{  }^{ 18 }{ C } } _{ r-1 } } +20\times { 2 }^{ 19 }$
$=20\times 19\times { 2 }^{ 18 }+20\times { 2 }^{ 19 }=20\times { 2 }^{ 18 }(19+2)=20\times 21\times { 2 }^{ 18 }=420\times { 2 }^{ 18 }\quad $
(3) option is correct

Multiple choice maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

The value of sin $ 2^o $ is approximately

  1. $ 2^o $
  2. $0.035$
  3. $ \frac {\pi}{180} $
  4. $0.017$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For small angles in radians, sin(x) is approximately x. 2 degrees = 2 * (pi/180) radians = pi/90 radians. pi/90 is approximately 3.14159 / 90 = 0.0349, which rounds to 0.035.

Multiple choice physics dynamics - explaining motion exponentiation index notation and products of prime factors powers

If 1 mg $ns^{-1}$ = $10^x \mu g ps^{-1}$, then the value of x is _________.

  1. 1

  2. 2

  3. -1

  4. 0
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
We know that
$1\ mg=10^3\ \mu g$
$1\ ns=10^3\ ps$

Therefore, it can be written as:
$1mgn{ s }^{ -1 }={ 10 }^{ 3 }\mu gn{ s }^{ -1 }$

$ =\dfrac { { 10 }^{ 3 } }{ { 10 }^{ 3 } } \mu gp{ s }^{ -1 }$

$ ={ 10 }^{ 0 }\mu gp{ s }^{ -1 }\\ \Rightarrow x=0$
Multiple choice mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

Value of $ \displaystyle \sin 45^{\circ} \cos 45 \left ( \tan 45^{\circ}+\cot 45^{\circ} \right )^{2}   $  is 

  1. $1$
  2. $2$
  3. $3$
  4. $4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$ \displaystyle \sin 45^{\circ} \cos 45 \left ( \tan 45^{\circ}+\cot 45^{\circ} \right )^{2}   $

$=\dfrac{1}{\sqrt2} \times \dfrac{1}{\sqrt2} (1+1)^2 $


$=\dfrac42$

$=2$

Multiple choice mathematics and statistics angle and their measurement angles and sides naming the sides in a right angled triangle understanding ratios

What is the value of $\sqrt {2}\sec 45^{\circ} - \tan 30^{\circ}$?

  1. $\dfrac {(2\sqrt {3} - 1)}{3}$
  2. $\dfrac {(\sqrt {3} - 1)}{\sqrt {3}}$
  3. $\dfrac {(2\sqrt {3} - 1)}{\sqrt {3}}$
  4. $\dfrac {(2\sqrt {3} + 1)}{3}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\sqrt {2}\sec 45^{\circ} - \tan 30^{\circ} = \sqrt {2}\times \sqrt {2} - \dfrac {1}{\sqrt {3}} = \dfrac {2\sqrt {3} - 1}{\sqrt {3}}$.

Multiple choice using trigonometric tables trigonometric ratios of some specific angles trigonometric identities trigonometry maths

$\left( 1-\dfrac { \cos { 61^{o} }  }{ \cos { 1^{o} }  }  \right) \left( 1-\dfrac { \cos { 62^{o} }  }{ \cos { 2^{o} }  }  \right) \left( 1-\dfrac { \cos { 63^{o} }  }{ \cos { 3^{o} }  }  \right) .......\left( 1-\dfrac { \cos { 119^{o} }  }{ \cos { 59^{o} }  }  \right) $

  1. $-1$
  2. $1$
  3. $2$
  4. $0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a telescoping product. Each term (1 - cos(60+x)/cos(x)) simplifies to (cos x - cos(60+x))/cos x = (2 sin(60/2 + x) sin(60/2))/cos x. The product eventually cancels out to -1.

Multiple choice using trigonometric tables trigonometric ratios of some specific angles trigonometric identities trigonometry maths

The value of $\tan \theta .\tan (\theta +60^{o})+\tan \theta \tan (\theta -60^{o})+\tan (\theta +60^{o}).\tan (\theta -60^{o})+3$ is  

  1. $0$
  2. $1$
  3. $-1$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the identity tan(A)tan(B)tan(C) = tan(A+B+C) - (tan A + tan B + tan C) or expanding the terms, the expression simplifies to 0.

Multiple choice using trigonometric tables trigonometric ratios of some specific angles trigonometric identities trigonometry maths

${\cos}^{2}{73}^{o}+{\cos}^{2}{47}^{o}+\cos{73}^{o}\cos{47}^{o}=.$

  1. $\dfrac{3}{4}$
  2. $-\dfrac{3}{4}$
  3. $\dfrac{4}{3}$
  4. $-\dfrac{4}{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using cos^2 A = (1 + cos 2A)/2, the expression becomes (1 + cos 146)/2 + (1 + cos 94)/2 + (cos 146 + cos 94)/2. This simplifies to 1 + (cos 146 + cos 94)/2 + (cos 146 + cos 94)/2 = 1 + cos 146 + cos 94. Using sum-to-product, this results in 3/4.

Multiple choice using trigonometric tables trigonometric ratios of some specific angles trigonometric identities trigonometry maths

$\dfrac { \cos{ 13 }^{ o }-\sin{ 13 }^{ o } }{ \cos{ 13 }^{ o }+\sin{ 13 }^{ o } } +\dfrac { 1 }{ \cot{ 148 }^{ o } }$ is equal to

  1. $1$
  2. $-1$
  3. $0$
  4. $\dfrac { 1 }{ 2 } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\dfrac{\cos 13 - \sin 13}{\cos 13 + \sin 13} + \dfrac{1}{\cot 148}$


$=\dfrac{\cos 13 (1 - \tan 13)}{\cos 13 (1 + \tan 13)} + \dfrac{1}{\cot (180 - 32)}$


$=\dfrac{\tan 45 - \tan 13}{1 + \tan 45 \tan 13} + \dfrac{1}{(-\cot 32)}$

$=\tan (45 - 13) - \tan 32$

$=\tan (32) - \tan 32$

$=0$