Tag: trigonometric ratios of acute angles

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Questions Related to trigonometric ratios of acute angles

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


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

  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. $2n\pi+\phi, n\in z$
Reveal answer Fill a bubble to check yourself
A Correct answer
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

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

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

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

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

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

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

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

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