Mathematics

Trigonometric Identities

82 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 using trigonometric tables trigonometric ratios of some specific angles trigonometric identities trigonometry maths

General solution of $cotx+tanx=2cosecx$ is

  1. $2n\pi\pm\dfrac{2\pi}{3},n\inZ$
  2. $2n\pi\pm\dfrac{4\pi}{3},n\in Z$
  3. $2n\pi\pm\dfrac{5\pi}{3},n\in Z$
  4. $2n\pi\pm\dfrac{\pi}{3},n\in Z$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Rewrite the equation cot(x) + tan(x) = 2 csc(x) in terms of sine and cosine: (cos(x)/sin(x)) + (sin(x)/cos(x)) = 2 / sin(x). Simplifying the left side gives 1 / (sin(x)cos(x)) = 2 / sin(x), which leads to cos(x) = 1/2 for sin(x) not equal to 0. Solving cos(x) = 1/2 yields x = 2n pi plus or minus pi/3.

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

$\tan 5\tan 25\tan 30\tan 65 \tan 85$ is equal to

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

Using complementary angle properties, tan(85) = cot(5) = 1/tan(5) and tan(65) = cot(25) = 1/tan(25). Thus, tan(5) and tan(85) cancel out, as do tan(25) and tan(65). We are left with tan(30 degrees), which equals 1 / sqrt(3).

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

$\tan 20 ^ { \circ } + \tan 40 ^ { \circ } + \sqrt { 3 } \tan 20 ^ { \circ } \tan 40 ^ { \circ }$  is equal to

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

From the identity tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B)), let A = 20 degrees and B = 40 degrees, so tan(60 degrees) = sqrt(3) = (tan(20) + tan(40)) / (1 - tan(20)tan(40)). Cross-multiplying gives tan(20) + tan(40) = sqrt(3) - sqrt(3)tan(20)tan(40), which rearranges to tan(20) + tan(40) + sqrt(3)tan(20)tan(40) = sqrt(3).

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}}).

Multiple choice

What is the value of (\tan 45^\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 45^\circ) is equal to (1).

Multiple choice

What is the value of (\tan 75^\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) for angles greater than (45^\circ). According to his table, (\tan 75^\circ) is equal to (\frac{\sqrt{6 + \sqrt{3}}}{\sqrt{6 - \sqrt{3}}}).

Multiple choice

What is the value of (\tan 90^\circ) according to Aryabhata?

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

Aryabhata did not define the value of (\tan 90^\circ). This is because (\tan \theta) is the ratio of the opposite side to the adjacent side, and in a right triangle with an angle of (90^\circ), the adjacent side is equal to (0). Therefore, (\tan 90^\circ) is undefined.

Multiple choice

What is the Pythagorean identity?

  1. sin^2(x) + cos^2(x) = 1

  2. sin(x) + cos(x) = 1

  3. tan(x) = sin(x) / cos(x)

  4. cot(x) = cos(x) / sin(x)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Pythagorean identity is a fundamental trigonometric identity that states that the square of the sine of an angle plus the square of the cosine of the same angle is equal to 1.