Tag: maths

Questions Related to maths

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

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

  1. $\dfrac{y}{r}$

  2. $\dfrac{r}{y}$

  3. $\dfrac{x}{r}$

  4. $\dfrac{r}{x}$

Show answer
Correct Option: D
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.

Find the exact value of sin $\theta$ 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. $\dfrac{4}{5}$

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

  3. $\dfrac{10}{5}$

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

Show answer
Correct Option: A
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.

Find the exact value of cosec $\theta$ 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. $\dfrac{5}{3}$

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

Show answer
Correct Option: C
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.

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

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

  1. $\dfrac{y}{r}$

  2. $\dfrac{x}{y}$

  3. $\dfrac{x}{r}$

  4. $\dfrac{r}{x}$

Show answer
Correct Option: B
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.

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

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

  1. $\dfrac{y}{r}$

  2. $\dfrac{r}{y}$

  3. $\dfrac{x}{r}$

  4. $\dfrac{r}{x}$

Show answer
Correct Option: B
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.


lf the distance between the points $(a \cos\theta, a \sin\theta), (a \cos\phi, a \mathrm{sin} \phi)$ is $2\mathrm{a}$ then $\theta=$.

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

  1. $2n\pi\pm\pi+\phi, n\in z$

  2. $n\displaystyle \pi\pm\frac{\pi}{2}+\phi, n\in z$

  3. $n\pi-\phi, n\in z$

  4. $2n\pi+\phi, n\in z$

Show answer
Correct Option: A
Explanation:
Given points $(a=cos\theta, a\sin\theta), (a\cos\phi, a\sin \phi)$

distance $2a$

distance between $(x _{1}, y _{1}), (x _{2}, y _{2})$

$\sqrt{(x _{2}-x _{1})^{2}+(y _{2}-y _{2})^{2}}$

So, 
$\sqrt{(a\cos\theta-a\cos\theta)^{2}+(a\sin\phi-a\sin\theta)^{2}}=2a$

Squaring on both sides

$\Rightarrow a^{2}(\cos^{2}\theta+\cos^{2}\phi-2\cos\theta\cos\phi)+a(\sin^{2}\phi+\sin^{2}\theta-2\sin\theta-\sin\phi)=4a^{2}$

$\Rightarrow a^{2}(\sin^{2}\theta+\cos^{2}\theta+\sin^{2}\theta+\cos^{2}\theta-2(\sin\theta-\sin\phi+\cos\theta\cos\phi))=4a^{2}$

$\Rightarrow a^{2}(1+1-2(\cos(\theta-\phi))=4a^{2}$

$\Rightarrow 2-2\cos(\theta-\phi)=4$

$\Rightarrow 2\cos(\theta-\phi)=-2$

$\Rightarrow \cos(\theta-\phi)=-1=\cos(\pi)$

$\Rightarrow \theta-\phi=2n\pi\pm \pi, n\in z$

$\Rightarrow \boxed{\theta=2n\pi\pm\pi+\phi, n\in z}$

Find the exact value of $\dfrac{\sin \theta}{\sec \theta}$ 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. $\dfrac{25}{12}$

  2. $\dfrac{12}{5}$

  3. $\dfrac{12}{25}$

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

Show answer
Correct Option: C
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.

A rectangle of length $4\ cm$ and breadth $3\ cm$ is scaled up $2$ times. What is the new length of the rectangle?

maths enlargement and scale drawing dilation enlargement similarity as a size transformation mapping mapping space around us bearing and drawings

  1. $6\ cm$

  2. $4\ cm$

  3. $8\ cm$

  4. $3\ cm$

Show answer
Correct Option: C
Explanation:

The size of the rectangle become double when it is scaled up $2$ times.

So new length $=2\times 4=8\ \ cm$
So option $C$ is correct.

A circle of radius $7\ cm$ is scaled $3$ times. Then the perimeter of the circle become:

maths enlargement and scale drawing dilation enlargement similarity as a size transformation mapping mapping space around us bearing and drawings

  1. $3$ times the original perimeter

  2. $6$ times the original perimeter

  3. $9$ times the original perimeter

  4. Doesn't change

Show answer
Correct Option: A
Explanation:

Radius of circle $=7 \ \ cm$

Perimeter $=2\pi r=2\times \pi\times7=14\pi\ \ cm$
When scaled $3$ times
New radius $=3\times 7=21 \ \ cm$
New Perimeter $=2\pi r=2\times \pi\times21=42\pi\ \ cm$
Ratio of perimeters $=\dfrac{42\pi}{14\pi}=3$
So the perimeter becomes three times.

If one shape becomes another using a resize, then the shapes are __________. 

maths enlargement and scale drawing dilation enlargement similarity as a size transformation mapping mapping space around us bearing and drawings

  1. similar

  2. congruent

  3. mirror images

  4. none of the above

Show answer
Correct Option: A
Explanation:

Resizing leads to change in scale factor and if the scale factor remains equal, then the figures will be similar to each other.