Mathematics

Trigonometric Identities and Equations

223 Questions

Solve a variety of questions based on trigonometric identities and mathematical equations. Key areas covered include double angle formulas, the law of sines, and quadrant angles. This material helps students preparing for advanced mathematics exams, UPSC, and state public service commissions.

Double angle formulasLaw of sinesTrigonometric quadrantsLaplace transformSecant functionTaylor series

Trigonometric Identities and Equations Questions

Multiple choice maths trigonometry trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities

Let $\theta$ be an angle in standard position with (x, y) a point on the terminal side of $\theta$ and r = $\sqrt{x^{2}+y^{2}}\neq 0$. What is cos $\theta$?

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

By using Pythagoras theorem, we will find the value of hypotenuse.
$x^{2}+y^{2}=r^{2}$
So, $\cos \theta$ = $\dfrac{adjacent \space\ side }{hypotenuse}$ = $\dfrac{y}{r}$

$\cos \theta$ = $\dfrac{x}{r}$

So, option C is correct.

Multiple choice maths trigonometry trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities

Find the exact value of sec $\theta$ trigonometric functions for the angle formed when the terminal side passes through (3, 4).

  1. $\dfrac{4}{5}$
  2. $\dfrac{3}{5}$
  3. $\dfrac{2}{5}$
  4. $\dfrac{5}{4}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

By using Pythagoras theorem, we will find the value of hypotenuse.
$3^{2}+4^{2}=c^{2}$
$9 + 16 = c^{2}$
$c = 5$
So, $\sec \theta$ = $\dfrac{hypotenuse}{adjacent \space\ side}$

$\sec \theta$ = $\dfrac{5}{4}$

So, option D is correct.

Multiple choice maths trigonometry trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities

Find sec $\theta$, if $\theta$ be an angle in standard position with (x, y) a point on the terminal side of $\theta$ and r = $\sqrt{x^{2}+y^{2}}\neq 0$.

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

By using Pythagoras theorem, we will find the value of hypotenuse.
$x^{2}+y^{2}=r^{2}$

So, sec $\theta$ = $\frac{hypotenuse}{adjacent\space\ side }$= $\dfrac{r}{x}$

sec $\theta$ = $\dfrac{r}{x}$

So, option D is correct.

Multiple choice maths trigonometry trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities

Find the exact value of sin $\theta$ trigonometric functions for the angle formed when the terminal side passes through (6, 8).

  1. $\dfrac{4}{5}$
  2. $\dfrac{6}{5}$
  3. $\dfrac{10}{5}$
  4. $\dfrac{3}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By using Pythagoras theorem, we will find the value of hypotenuse.
$6^{2}+8^{2}=c^{2}$
$36 + 64 = c^{2}$
$c = 10$
So, $\sin \theta$ = $\dfrac{opposite \space\ side }{hypotenuse}$

= $\dfrac{8}{10}$ = $\dfrac{4}{5}$

So, option A is correct.

Multiple choice maths trigonometry trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities

Find the exact value of cosec $\theta$ trigonometric functions for the angle formed when the terminal side passes through $(3, 4).$

  1. $\dfrac{4}{5}$
  2. $\dfrac{3}{5}$
  3. $\dfrac{5}{3}$
  4. $\dfrac{5}{4}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By using Pythagoras theorem, we will find the value of hypotenuse.
$3^{2}+4^{2}=c^{2}$
$9 + 16 = c^{2}$
$c = 5$
So, $\csc \theta$ = $\dfrac{hypotenuse}{opposite \space\ side}$

$\csc \theta$ = $\dfrac{5}{3}$

So, option C is correct.

Multiple choice maths trigonometry trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities

Find cot $\theta$, if $\theta$ be an angle in standard position with (x, y) a point on the terminal side of $\theta$ and r = $\sqrt{x^{2}+y^{2}}\neq 0$.

  1. $\dfrac{y}{r}$
  2. $\dfrac{x}{y}$
  3. $\dfrac{x}{r}$
  4. $\dfrac{r}{x}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

By using Pythagoras theorem, we will find the value of hypotenuse.
$x^{2}+y^{2}=r^{2}$
So, cot $\theta$ = $\dfrac{adjacent\space\ side}{opposite\space\ side}$= $\dfrac{x}{y}$


cot $\theta$ = $\dfrac{x}{y}$

So, option B is correct.

Multiple choice maths trigonometry trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities

Find cosec $\theta$, if $\theta$ be an angle in standard position with (x, y) a point on the terminal side of $\theta$ and r = $\sqrt{x^{2}+y^{2}}\neq 0$.

  1. $\dfrac{y}{r}$
  2. $\dfrac{r}{y}$
  3. $\dfrac{x}{r}$
  4. $\dfrac{r}{x}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

By using Pythagoras theorem, we will find the value of hypotenuse.
$x^{2}+y^{2}=r^{2}$
So, cosec $\theta$ = $\dfrac{hypotenuse}{opposite\space\ side }$= $\dfrac{r}{y}$

cosec $\theta$ = $\dfrac{r}{y}$

So, option B is correct.

Multiple choice maths trigonometry trigonometric ratios of acute angles compound angles, multiple angles, sub multiple angles and transformation formulae trigonometric identities

Find the exact value of $\dfrac{\sin \theta}{\sec \theta}$ trigonometric functions for the angle formed when the terminal side passes through $(3, 4)$.

  1. $\dfrac{25}{12}$
  2. $\dfrac{12}{5}$
  3. $\dfrac{12}{25}$
  4. $\dfrac{5}{4}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By using Pythagoras theorem, we will find the value of hypotenuse.
$3^{2}+4^{2}=c^{2}$
$9 + 16 = c^{2}$
$c = 5$

So, $\dfrac{\sin\theta}{\sec\theta}$ = $\dfrac{opposite \space\ side \times adjacent \space side}{hypotenuse \times hypotenuse}$
= $\dfrac{3\times 4}{5\times 5}$
=$\dfrac{12}{25}$

So, option C is correct.

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

If $A+B=\dfrac { \pi  }{ 3 } $ and $\cos { A } +\cos { B } =1 $, then which of the following are true: 

  1. $\cos { \left( A-B \right) =\dfrac { 1 }{ 3 } } $
  2. $\cos { \left( A-B \right) =-\dfrac { 1 }{ 3 } } $
  3. $\left| \cos { A } -\cos { B } \right| =\sqrt { 2/3 } $
  4. $\left| \cos { A } -\cos { B } \right| =\cfrac { 1 }{ \sqrt { 3 } } $
Reveal answer Fill a bubble to check yourself
B,C Correct answer
Explanation

Ans. $(b)$, $(c)$

From the given relation, we have 
$2\cos { \cfrac { A+B }{ 2 }  } \cos { \cfrac { A-B }{ 2 } =1 }$
Or $2\cos { {30}^{o} } \cos { \cfrac { A-B }{ 2 }  } =1$
$\therefore\quad \cos { \cfrac { A-B }{ 2 }  } =\cfrac { 1 }{ \sqrt { 3 }  }$
$\therefore\quad \cos { \left( A-B \right)  } =\cos ^{ 2 }{ \cfrac { A-B }{ 2 } -1 } =2.\cfrac { 1 }{ 3 } -1=-\cfrac { 1 }{ 3 } \Rightarrow \left( b \right)$
Again $\left| \cos { A } -\cos { B }  \right| =2\sin { \cfrac { A+B }{ 2 }  } \sin { \cfrac { B-A }{ 2 }  }$ 
$=2\sin { {30}^{o} } \sqrt { 1-\cos ^{ 2 }{ \cfrac { A-B }{ 2 }  }  } =1\sqrt { 1-\cfrac { 1 }{ 3 }  } =\sqrt { \cfrac { 2 }{ 3 }  }$

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 trigonometric equations trigonometric functions trigonometry maths

$\alpha, \beta$ are the solution (s) of $3 cos 2 \theta + 4 sin 2 \theta = 5$
$tan (\alpha + \beta) = $

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

The equation 3cos(2theta) + 4sin(2theta) = 5 can be written as 5(3/5 cos(2theta) + 4/5 sin(2theta)) = 5, or cos(2theta - phi) = 1, where tan(phi) = 4/3. The solutions alpha and beta lead to tan(alpha + beta) = 1.

Multiple choice trigonometric equations trigonometric functions trigonometry maths

$\alpha, \beta$ are the solution (s) of $3 cos 2 \theta + 4 sin 2 \theta = 5$
$tan (\alpha - \beta) = $

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

Since 3cos(2theta) + 4sin(2theta) = 5 has only one solution for 2theta in the range [0, 2pi), alpha and beta are essentially the same value (or differ by a multiple of pi), making tan(alpha - beta) = 0.

Multiple choice trigonometric equations trigonometric functions trigonometry maths

If $\alpha$ is the angle of first quadrant such that $co\sec ^{ 4 }{ \alpha  }=17+\cot ^{ 4 }{ \alpha  } $, then what is the value of $\sin{\alpha}$?

  1. $\cfrac{1}{3}$
  2. $\cfrac{1}{4}$
  3. $\cfrac{1}{9}$
  4. $\cfrac{1}{16}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$ cosec^{4}\alpha -cot^{4}\alpha = 17$

 (As $ cosec^{2}\alpha -cot^{2}\alpha=1) $

$ \Rightarrow (cosec^{2}\alpha -cot^{2}\alpha )(cosec^{2}\alpha +cot^{2}\alpha ) = 17 $ 

$ \Rightarrow cosec^{2}\alpha +cot^{2}\alpha = 17...(1) $

$ cosec^{2}\alpha -cot^{2}\alpha = 1...(2) $

then $ (1) + (2) \Rightarrow 2cosec^{2}\alpha = 18 $

$ \Rightarrow sin^{2}\alpha = \dfrac{1}{9}\Rightarrow \boxed{sin\,\alpha = \dfrac{1}{3}} $ $ \left ( \because \alpha \,in\,1st\,quadrant \right ) $