Tag: trigonometry

Quizzes (11) Questions (86)

Questions Related to trigonometry

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

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

  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
Reveal answer Fill a bubble to check yourself
C Correct answer
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}}$ 

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

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 $B$?

  1. $160km$
  2. $180km$
  3. $200km$
  4. $220km$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 

$\Rightarrow$ $x = 180 km.$

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

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.

  1. ${ 55 }^{ o },\quad { 28 }^{ o }\quad \& \quad { 97 }^{ o }\\$
  2. ${ 65 }^{ o },\quad { 22 }^{ o }\quad \& \quad { 93 }^{ o }\\$
  3. $\\{ 60 }^{ o },\quad { 24 }^{ o }\quad \& \quad { 96 }^{ o }\\$
  4. ${ 70 }^{ o },\quad { 15 }^{ o }\quad \& \quad { 95 }^{ o }\\$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Convert all angles to degrees: (2x/3) grades = (2x/3) * (9/10) = 0.6x degrees. (3x/2) degrees = 1.5x degrees. (2*pi*x/75) radians = (2*pi*x/75) * (180/pi) = 4.8x degrees. Sum = 0.6x + 1.5x + 4.8x = 6.9x = 180. x = 180/6.9 = 26.08. Checking the options, C gives 60, 24, 96 which sum to 180.

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

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

  1. $a + b$
  2. $\displaystyle \sqrt{ab}$
  3. $\displaystyle a\times b$
  4. $\displaystyle a\sqrt{b}$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $\displaystyle h=\sqrt{ab}$

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.