Mathematics

Trigonometric Identities

87 Questions

Trigonometric identities focus on solving equations using tangent, sine, and cosine properties. Questions involve half angle formulas and slope calculations. This topic is a staple in the quantitative aptitude section of major competitive examinations.

Tangent propertiesAngle formulasTrigonometric equationsHalf angle identitiesSlope calculations

Trigonometric Identities Questions

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

If $\tan A = \dfrac {1 - \cos B}{\sin B}$, then the value of $\dfrac {2\tan A}{1 - \tan^{2}A}$ is

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

Given, $\tan A =  \dfrac {1 - \cos B}{\sin B} $


                       $= \dfrac {2\sin^{2}\dfrac {B}{2}}{2\sin \dfrac {B}{2}\cdot \cos \dfrac {B}{2}}$


                       $= \tan \dfrac {B}{2}$

Therefore, $A = \dfrac {B}{2} \Rightarrow 2A = B$

Now $\dfrac {2\tan A}{1 - \tan^{2}A} = \tan 2A = \tan B$

Multiple choice maths binomial theorem, sequence and series series introduction to series introduction to sequences and series

If  in traingle ABC $\cos 2B=\dfrac {\cos (A+C)}{\cos (A-C)}$, then 

  1. $\tan A, \tan B, \tan C$ are in $A.P$
  2. $\tan A, \tan B, \tan C$ are in $G.P$
  3. $\tan A, \tan B, \tan C$ are in $H.P$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given cos(2B) = cos(A+C)/cos(A-C). Using componendo and dividendo, (1-cos(2B))/(1+cos(2B)) = (cos(A-C)-cos(A+C))/(cos(A-C)+cos(A+C)). This simplifies to tan^2(B) = tan(A)tan(C), meaning tan(A), tan(B), tan(C) are in G.P.

Multiple choice physics trigonometrical ratios angles and sides naming the sides in a right angled triangle angle and their measurement

If $E. \ tan(x -
30^{\circ}) = j. \ tan(x+120^{\circ})$, then $\frac{E + J}{E-J} =$

  1. $\ sin 2x$
  2. $2 \ cos 2x$
  3. $\ tan2x$

  4. None of these.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given $E\tan (x-30^{\circ})=J\tan (x+120^{\circ})$

$\implies \dfrac{E}{J}=\dfrac{\tan (x+120^{\circ})}{\tan (x-30^{\circ})}$

Applying compoundo and dividendo rule

$\dfrac{E+J}{E-J}=\dfrac{\tan (x-30^{\circ})+\tan (x+120^{\circ})}{\tan (x+120^{\circ})-\tan (x-30^{\circ})}=\dfrac{\frac{\sin (x-30^{\circ})}{\cos (x-30^{\circ})}+\frac{\sin (x+120^{\circ})}{\cos (x+120^{\circ})}}{\frac{\sin (x+120^{\circ})}{\cos (x+120^{\circ})}-\frac{\sin (x-30^{\circ})}{\cos (x-30^{\circ})}}$

                                                        $=\dfrac{\sin (x-30^{\circ})\cos(x+120^{\circ})+\sin (x+120^{\circ})\cos(x-30^{\circ})}{\sin (x+120^{\circ})\cos(x-30^{\circ})-\sin (x-30^{\circ})\cos (x+120^{\circ})}$

                                                       $=\dfrac{\sin (x-30^{\circ}+x+120^{\circ})}{\sin (x+120^{\circ}-x+30^{\circ})}$

                                                      $=\dfrac{\sin (90^{\circ}+2 x)}{\sin 150^{\circ}}$

                                                      $=2\cos 2 x$
Multiple choice using trigonometric tables trigonometric ratios of some specific angles trigonometric identities trigonometry maths

The value of $\tan 7\dfrac{1}{2}^{o}$ is equal to

  1. $\sqrt{6}+\sqrt{3}+\sqrt{2}-2$
  2. $\sqrt{6}-\sqrt{3}+\sqrt{2}-2$
  3. $\sqrt{6}-\sqrt{3}+\sqrt{2}+2$
  4. $\sqrt{6}-\sqrt{3}-\sqrt{2}-2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using the formula tan(x/2) = sqrt((1-cos x)/(1+cos x)) with x = 15 degrees, or using the half-angle identity for 7.5 degrees, the value is sqrt(6) + sqrt(3) + sqrt(2) - 2.

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

The value of $\sqrt { 3 } tan{ 10 }^{ 0 }+\sqrt { 3 } tan{ 20 }^{ 0 }+tan{ 10 }^{ 0 }tan{ 20 }^{ 0 }$ is ___________.

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

$\tan (30) = \tan (20 + 10)$


$\dfrac{1}{\sqrt{3}} = \tan 30 = \dfrac{\tan 20 + \tan 10}{1 - \tan 20 \tan 10}$


$1 - \tan 20 \tan 10 = \sqrt{3} (\tan 20 + \tan 10)$

$\sqrt{3} \tan 20 + \sqrt{3} \tan 10 + \tan 10 \tan 20 = 1$

Multiple choice physics parametric equations proving properties of curves derivatives - introduction and interpretation introduction to calculus - differentiation

If $x = \exp \left{ \tan ^ { - 1 } \left( \frac { y - x ^ { 2 } } { x ^ { 2 } } \right) \right}$ then $\frac { d y } { d x } =$

  1. $2 x [ 1 + \tan ( \log x ) ] + x \cdot \sec ^ { 2 } ( \log x )$
  2. $x [ 1 + \tan ( \log x ) ] + \sec ^ { 2 } ( \log x )$
  3. $2 x [ 1 + \tan ( \log x ) ] + x ^ { 2 } \sec ^ { 2 } ( \log x )$

  4. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$x=exp\left\{\tan^{-1}\left(\dfrac{y-x^2}{x^2}\right)\right\}$
$ln x=\tan^{-1}\left(\dfrac{y-x^2}{x^2}\right)$
$\Rightarrow \tan (ln x)=\dfrac{y}{x^2}-1$
$\Rightarrow y=x^2[\tan (ln x)+1]$
$=x^2\tan(ln x)+x^2$
$\dfrac{dy}{dx}=2x\tan (ln x)+x^2\sec^2(ln x)\dfrac{1}{x}+2x$
$=2x\tan (ln x)+x\sec^2(ln x)+2x$.
Multiple choice maths algebraic functions, equations and inequalities descartes rule sign of quadratic expression sum and product of the roots of a polynomial equation

The general solution of the equation 
$tan \, x + tan \, 2x + \sqrt{3} \, tan \, x \, tan \, 2x = \sqrt{3}$ is 

  1. $x = \dfrac{n \pi}{3} + \dfrac{\pi}{9}, \, n \in z$
  2. $x = m \pi + \dfrac{\pi}{9}, \, n \in z$
  3. $x = \dfrac{(n + 1) \pi}{3}, n \in z$
  4. $x = \dfrac{n \pi}{3} + \dfrac{\pi}{3}, \, n \in z$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\tan x+\tan 2x+\sqrt{3}\tan x\, \tan 2x=\sqrt{3}$

$\tan x+\tan 2x=\sqrt{3}-\sqrt{3}\tan x\, \tan 2x$

$\cfrac{\tan x+\tan 2x}{1-\tan x\, \tan 2x}=\sqrt{3}$

$\tan (x+2x)=\sqrt{3}$

$\tan 3x=\sqrt{3}$

$\tan 3x=\tan \cfrac{\pi }{3}$

$\Rightarrow 3x=n\pi+ \cfrac{\pi }{3}$   ( where $n$ is an integer )

$x=\cfrac{n\pi }{3}+\cfrac{\pi }{9}$
Multiple choice maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

If the non-zero terms $x , y, z$ are in $AP $ and $\tan ^ { -1 } x, \tan ^ { - 1 } y, \tan ^ { - 1 } x$ are also $AP$ then

  1. $x = y = z$
  2. $n ^ { 2 } y = z$
  3. $z ^ { 2 } = x y$
  4. $y ^ { 2 } = \frac { 1 } { x 2 }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given $x,yz$ are in $AP$

So, $y=x=2-y$
$2y=x+2 \quad -(1)$
Similarly,
$\tan ^{ -1 }{ x } ,\tan ^{ -1 }{ y } $ and $\tan ^{ -1 }{ z } $ are also in $AP$.
So, $2\tan ^{ -1 }{ y } =\tan ^{ -1 }{ x } +\tan ^{ -1 }{ z } $
$\Rightarrow \tan ^{ -1 }{ (\cfrac { 2y }{ 1-{ y }^{ 2 } } ) } =\tan ^{ -1 }{ (x+\cfrac { z }{ 1-xz } ) } \ \Rightarrow \cfrac { 2y }{ 1-{ y }^{ 2 } } =x+\cfrac { z }{ 1-xz } \ \Rightarrow x+\cfrac { z }{ 1-{ y }^{ 2 } } =x+\cfrac { z }{ 1-xz } \ \Rightarrow \cfrac { 1 }{ 1-{ y }^{ 2 } } =\cfrac { 1 }{ 1-xz } \ \Rightarrow 1-xz=1-{ y }^{ 2 }\ \Rightarrow 1-xz=1-{ { \cfrac { x+z }{ 2 } }  }^{ 2 }\ \Rightarrow { x }^{ 2 }+{ z }^{ 2 }+2xz=4xz\ \Rightarrow { x }^{ 2 }+{ z }^{ 2 }+2xz-4xz=0\ \Rightarrow { x }^{ 2 }+{ z }^{ 2 }-2xz=0\ \Rightarrow { (x-z) }^{ 2 }=0$
$\Rightarrow x=z$ put in $(1)$
$\Rightarrow 2y=x+2\ \Rightarrow 2y=2x\ \Rightarrow y=x\ x=y=z$

Multiple choice

What is the value of (\tan 30^\circ) according to Aryabhata?

  1. \(\frac{1}{2}\)
  2. \(\frac{\sqrt{3}}{2}\)
  3. \(\frac{1}{\sqrt{2}}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

Aryabhata used the definition of (\tan \theta) as (\frac{\sin \theta}{\cos \theta}) to calculate the values of (\tan \theta). According to his table, (\tan 30^\circ) is equal to (\frac{1}{\sqrt{3}}).