Tag: trigonometric ratios of acute angles

Questions Related to trigonometric ratios of acute angles

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

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

  1. True

  2. False

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Correct Option: B
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 $\text{False}$

If $tan x + cot x = 2$, then $sin^{2n}x+cos^{2n}x=$

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

  1. $\dfrac{1}{2}$

  2. $2^n$

  3. $\dfrac{1}{2^n}$

  4. $\dfrac{1}{2^{n-1}}$

Show answer
Correct Option: D
Explanation:

Given $\tan x+\cot x=2$

$\implies \tan x+\dfrac{1}{\tan x}=2$
$\implies \tan^2 x-2\tan x+1=0$
$\implies (\tan x-1)^2=0$
$\implies \tan x=1\implies x=\dfrac{\pi}{4}$
$\sin^{2 n} x+\cos^{2 n} x=\bigg(\dfrac{1}{\sqrt{2}}\bigg)^{2 n}+\bigg(\dfrac{1}{\sqrt{2}}\bigg)^{2 n}=\dfrac{1}{2^n}+\dfrac{1}{2^n}=\dfrac{2}{2^n}=\dfrac{1}{2^{n-1}}$

If $\tan x =\dfrac{3}{4} , \pi < x < \dfrac{3\pi}{2} $ find value of $\sin\dfrac{x}{2} , \cos\dfrac{x}{2} , \tan \dfrac{x}{2}$

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

  1. cos x/2= 3/2, sin x/2 =-3/2, tan x/2= -1/√10

  2. cos x/2= 3/√10, sin x/2 =-3/2, tan x/2= -1/√10

  3. cos x/2= -1/√10, sin x/2 = 3/√10, tan x/2= -3

  4. cos x/2= 3/√10, sin x/2 =-2/3, tan x/2= -1/√10

Show answer
Correct Option: C
Explanation:
Given 

$ \pi < x < \dfrac {3\pi} 2$ 

$ \implies \dfrac \pi 2 < \dfrac x2 < \dfrac {3\pi }4 $ 

$ \implies \cos \dfrac x2 <0 , \sin \dfrac x2 >0 $ 

$ \tan x =\dfrac 34 $ 

$ \dfrac {2 \tan \dfrac x2 }{1-\tan ^2 \dfrac x2 }=\dfrac 34$

$ 8 \tan \dfrac x2 =3-3\tan ^2 \dfrac x2 $

$ 3\tan ^2 \dfrac x2 +8\tan \dfrac x2 -3=0 $ 

$ 3\tan ^2 \dfrac x2 +9\tan \dfrac x2 - \left(\tan \dfrac x2 +3 \right)=0 $ 

$ \tan \dfrac x2 =-3,\dfrac 13 $ 

As $ tan \dfrac x2 <0 \implies \tan \dfrac x2 =-3$ 

$ \cos \dfrac x2 =\dfrac {-1}{\sqrt {(-1)^2+3^2}}=\dfrac {-1}{\sqrt {10}}$ 

$ \sin \dfrac x2 =\dfrac {3}{\sqrt {(-1)^2+3^2}}=\dfrac {3}{\sqrt {10}}$ 

A boat takes $19$ hours for travelling downstream from point $A$ to point $B$ and coming back to a point $C$ midway between $A$ and $B$. If the velocity of the stream is $4$ $kmph$ and the speed of the boat in still water is $14 \mathrm { kmph}, $ what is the distance between $A$ and $B$?

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

  1. $160km$

  2. $180km$

  3. $200km$

  4. $220km$

Show answer
Correct Option: B
Explanation:


Speed in downstream =  $(14 + 4) km/hr = 18 km/hr ;$

Speed in upstream =  $(14 – 4) km/hr = 10 km/hr. $

Let the distance between A and B be $x$ km. Then,

$\frac{x}{18} + \frac{x}{2}\times\frac{1}{10}$ = 19 

$\Rightarrow \frac{x}{18} + \frac{x}{20}$ = 19 

$\Rightarrow$ $x = 180 km.$

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

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

  1. $n\ \pi \pm \dfrac {\pi}{4}$

  2. $2n\ \pi \pm \dfrac {\pi}{4}$

  3. $2n\ \pi \pm \dfrac {\pi}{4}$

  4. $\dfrac {n\pi}{2}+(-1)^{n} ,\dfrac {\pi}{4}$

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Correct Option: A

$\sin\ (45^{o}+\theta)-\cos\ (45^{o}-\theta)$ is equal to

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

  1. $2\cos \theta$

  2. $0$

  3. $2\sin \theta$

  4. $1$

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Correct Option: A

One angle of a triangle is $\displaystyle \frac{2x}{3}$ grades another is $\displaystyle \frac{3x}{2}$ degrees, whilst the third is $\displaystyle \frac{2\pi x}{75}$ radians ; express them all in degrees.

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

  1. ${ 55 }^{ o },\quad { 28 }^{ o }\quad &amp; \quad { 97 }^{ o }\$

  2. ${ 65 }^{ o },\quad { 22 }^{ o }\quad &amp; \quad { 93 }^{ o }\$

  3. $\{ 60 }^{ o },\quad { 24 }^{ o }\quad &amp; \quad { 96 }^{ o }\$

  4. ${ 70 }^{ o },\quad { 15 }^{ o }\quad &amp; \quad { 95 }^{ o }\$

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Correct Option: C

The angles of elevation of the top of the tower from two points at distances '$a$' and '$b$' from the base and in the same straight line with it complementary The height of the tower is

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

  1. $a + b$

  2. $\displaystyle \sqrt{ab}$

  3. $\displaystyle a\times b$

  4. $\displaystyle a\sqrt{b}$

Show answer
Correct Option: B
Explanation:

AP=a,AQ=b
$\displaystyle \tan \theta =\frac{h}{a}$....(i)
$\displaystyle \tan \left ( 90-\theta  \right )=\frac{h}{b}$....(ii)
$\displaystyle \Rightarrow \cot \theta =\frac{h}{b}$
$\displaystyle \Rightarrow \tan \theta \times \cot \theta = \frac{h}{a}\times \frac{h}{b}=1$
$\displaystyle \Rightarrow h^{2}=ab$
or $\displaystyle h=\sqrt{ab}$

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

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

  1. $\dfrac{3}{5}$

  2. $\dfrac{6}{5}$

  3. $\dfrac{4}{5}$

  4. $\dfrac{8}{5}$

Show answer
Correct Option: C
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.

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

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

  1. $\dfrac{4}{5}$

  2. $\dfrac{3}{5}$

  3. $\dfrac{2}{5}$

  4. $\dfrac{1}{5}$

Show answer
Correct Option: A
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.