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

sin90=?

  1. 0

  2. 1

  3. 2

  4. 3

  5. 4
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

sin90=1

Multiple choice

A series expansion for the function sin$\theta$ is
  1. 1 - $\frac{\theta^2}{2!} + \frac{\theta^4}{4!}$
  2. $\frac{\theta^3}{3!} + \frac{\theta^5}{5!}$
  3. 1 + $\theta$+ $\frac{\theta^2}{2!} + \frac{\theta^3}{3!}$
  4. $\theta$ + $\frac{\theta^3}{3!} + \frac{\theta^5}{5!}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\text{We know the series expansion of} \\ sin \theta = \theta - \frac{\theta^3}{\lfloor{3}} + \frac{\theta^3}{\lfloor{5}} - \frac{\theta^7}{\lfloor{7}}+ ......$

Multiple choice maths complementary angle, supplementary angles and adjcent angles acute and obtuse angles types of angle measurement of an angle

In $\Delta ABC\,,\,if\,\,A\,\,:\,\,B\,:\,\,C\, = \,1\,\,:\,\,5\,\,:\,\,6\,\,then$ find the value of $\sin A: \sin B: \sin C$

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

Given that $ A:B:C = 1:5:6 $


We know that the sum of angles in a triangle is $180^0$

Let us first find each angle.

Total no.of parts= $1+5+6= 12$

$A$= $ \dfrac{1}{12} $ ($180^0$) =$15^0$

$B$= $ \dfrac{5}{12} $ ($180^0$) =$75^0$

$C$= $ \dfrac{6}{12} $ ($180^0$) =$90^0$

Hence, $ \sin A: \sin B: \sin C$ = $\sin 15^o$$:\sin 75^o$$:\sin 90^o$

$=\dfrac {\sqrt{3}-1} {2\sqrt 2} : \ \dfrac{\sqrt3+1}{2\sqrt2 } :1$ 

$= ({\sqrt{3}-1})  : ({\sqrt3+1} ): ({2\sqrt2 }) $ 

Multiple choice validating statements proofs in mathematics mathematical reasoning maths

While simplifying $\sqrt { \frac { 1-cosx }{ 1+cosx }  } $, two students got the following two answers A & B.
A)  cosec x - cot x  (B)   $\frac { 1 }{ cosecx+cotx } $ What can you say about answers ?

  1. Both A & B are worng

  2. Both A & B are right

  3. A is right B is wrong

  4. B is right A is wrong
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths complex numbers and linear inequations argand plane and polar representation geometric representation of a complex number complex numbers and quadratic equations

If $\sin \frac {6\pi}5+i(1+\cos \frac {6\pi }5)$ then

  1. $|Z|=-2\cos \frac {3\pi}5$
  2. $Arg(Z)=\frac {\pi}5$
  3. $Arg(Z)=\frac {9\pi }{10}$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Z = sin(6pi/5) + i(1 + cos(6pi/5)). Using half-angle identities: sin(6pi/5) = 2sin(3pi/5)cos(3pi/5) and 1+cos(6pi/5) = 2cos^2(3pi/5). Factoring out 2cos(3pi/5) gives Z = 2cos(3pi/5) * (sin(3pi/5) + i cos(3pi/5)). The argument is 9pi/10.

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

$\sin ^{ 8 }{ \theta  } -\cos ^{ 8 }{ \theta  } -\left( \sin ^{ 2 }{ \theta  } -\cos ^{ 2 }{ \theta  }  \right) \left( 1-\sin ^{ 2 }{ \theta  }  \right) $=0

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\sin^4 \theta+\cos^4 \theta=(\sin^2 \theta+\cos^2 \theta)^2-2\sin^2 \theta\cos^2 \theta=1-2\sin^2 \theta\cos^2 \theta$

$\sin^4 \theta-\cos^4 \theta=(\sin^2 \theta-\cos^2 \theta)(\sin^2 \theta+\cos^2 \theta)=\sin^2 \theta-\cos^2 \theta$
$\sin^8\theta-\cos^8\theta-(\sin^2 \theta-\cos^2 \theta)(1-\sin^2 \theta)=(\sin^4 \theta+\cos^4 \theta)(\sin^4 \theta-\cos^4 \theta)-(\sin^2 \theta-\cos^2 \theta)(1-\sin^2 \theta)$
                                                                                 $=(1-2\sin^2 \theta\cos^2 \theta)(\sin^2 \theta-\cos^2 \theta)-(\sin ^2 \theta-\cos^2\theta)(1-\sin^2 \theta)$
                                                                                 $=(\sin^2 \theta-\cos^2 \theta)(\sin^2 \theta)(1-2\cos^2 \theta)$
So the given relation is $\text{False}$

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

If $\tan \theta+\left(\dfrac {\pi}{2}+\theta\right)=0$ then the most general value of $\theta$ is (where $n\ \in\ Z$)

  1. $n\ \pi \pm \dfrac {\pi}{4}$
  2. $2n\ \pi \pm \dfrac {\pi}{4}$
  3. $2n\ \pi \pm \dfrac {\pi}{4}$
  4. $\dfrac {n\pi}{2}+(-1)^{n} ,\dfrac {\pi}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation is tan(theta) + tan(pi/2 + theta) = 0. Since tan(pi/2 + theta) = -cot(theta), we have tan(theta) - cot(theta) = 0, so tan(theta) = cot(theta). This implies tan^2(theta) = 1, so tan(theta) = +/- 1. Thus, theta = n*pi +/- pi/4.

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

What is the exact value of $cos \theta$ trigonometric functions for the angle formed when the terminal side passes through $(6, 8)$?

  1. $\dfrac{3}{5}$
  2. $\dfrac{6}{5}$
  3. $\dfrac{4}{5}$
  4. $\dfrac{8}{5}$
Reveal answer Fill a bubble to check yourself
C 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, $\cos \theta$ = $\dfrac{opposite \space\ side }{hypotenuse}$ = $\dfrac{8}{10}$

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

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

Determine the exact value of $\cos \theta$ 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. $\dfrac{1}{5}$
Reveal answer Fill a bubble to check yourself
A 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, $\cos\theta$ = $\dfrac{adjacent \space\ side}{hypotenuse}$

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

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

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.