Tag: trigonometry

Quizzes (11) Questions (86)

Questions Related to trigonometry

Multiple choice trigonometric equations trigonometric functions trigonometry maths

$tan  5x-tan  3x-tan  2x=$

  1. $\tan 5x \tan 3x \tan 2x$
  2. $\sin 5x \sin 3x \sin 2x$
  3. $\cos 5x \cos 3x \cos 2x$
  4. $\sec 5x \sec 3x \sec 2x$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We've,

$\tan (3x+2x)=\tan 5x$

or, $\dfrac{\tan 3x+\tan 2x}{1-\tan 3x.\tan 2x}=\tan 5x$

or, $\tan 3x+\tan 2x=\tan 5x-\tan 2x.\tan 3x.\tan 5x$

or, $\tan 5x-\tan 3x-\tan 2x=\tan 2x.\tan 3x.\tan 5x$.

Multiple choice trigonometric equations trigonometric functions trigonometry maths

$A, B, C$ are three angles such that $\tan  A+\tan  B+\tan  C=\tan  A  \tan  B  \tan  C.$ Which of the following statements is always correct ?

  1. $ABC$ is a triangle, i.e. $A+B+C=\pi $
  2. $A=B=C. i.e., $ $ABC$ is an equilateral triangle
  3. $A+B=C, $ i.e., $ABC$ is a right- angled triangle
  4. $A+B=\pi $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

(A) $tan\left [ (A+B)+C \right ]$
$=\frac{tan (A+B)+tan C}{1-tan (A+B)  tan  C}=\frac{\frac{tan  A+tan   B}{1-tan  A   tan  B}+tan  C}{1-\frac{tan  A+tan  B}{1-tan   A   tan  B}.  tan  C}$
$=\frac{tan  A+tan  B+tan  C-tan  A   tan  B   tan  C}{Denominator}$
$=0$
$\left [ since,  tan  A+tan  B+tan  C =tan  A   tan  B   tan  C \right ]$
$\therefore A+B+C=\pi $ i.e.,  A, B, C is a triangle

Multiple choice trigonometric equations trigonometric functions trigonometry maths

If $\dfrac{\pi}{4}<A<\dfrac{\pi}{2}$ then $\tan^{-1}\left(\dfrac{1}{2}\tan 2A\right)+\tan^{-1}(\cot A)+\tan^{-1}(\cot^{3}A)$=

  1. $0$
  2. $\pi$
  3. $\pi/2$
  4. $\pi/4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using trigonometric identities for inverse functions and the given range, the sum simplifies to 0. Specifically, the terms cancel out based on the properties of inverse tangents of cotangent functions.

Multiple choice trigonometric equations trigonometric functions trigonometry maths

If $A+B+C=\pi $ and cosA=cosB cosC, then tanB tanC is equal to 

  1. $\frac { 1 }{ 2 } $
  2. $2$
  3. $1$
  4. $-\frac { 1 }{ 2 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given $A+B+C=\pi\implies A=\pi-(B+C)$

And also given $\cos A=\cos B\cos C$
$\implies \cos (\pi-(B+C))=\cos B\cos C$
$\implies -\cos B\cos C+\sin B\sin C=\cos B\cos C$
$\implies \sin B\sin C=2\cos B\cos C$
$\implies \tan B\tan C=2$

Multiple choice trigonometric equations trigonometric functions trigonometry maths

$\alpha, \beta$ are the solution (s) of $3 cos 2 \theta + 4 sin 2 \theta = 5$
$tan (\alpha + \beta) = $

  1. $1$
  2. $\dfrac{3}{4}$
  3. $\dfrac{4}{3}$
  4. $\dfrac{1}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation 3cos(2theta) + 4sin(2theta) = 5 can be written as 5(3/5 cos(2theta) + 4/5 sin(2theta)) = 5, or cos(2theta - phi) = 1, where tan(phi) = 4/3. The solutions alpha and beta lead to tan(alpha + beta) = 1.

Multiple choice trigonometric equations trigonometric functions trigonometry maths

$\alpha, \beta$ are the solution (s) of $3 cos 2 \theta + 4 sin 2 \theta = 5$
$tan (\alpha - \beta) = $

  1. $0$
  2. $1$
  3. $\dfrac{1}{4}$
  4. $\dfrac{4}{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since 3cos(2theta) + 4sin(2theta) = 5 has only one solution for 2theta in the range [0, 2pi), alpha and beta are essentially the same value (or differ by a multiple of pi), making tan(alpha - beta) = 0.

Multiple choice trigonometric equations trigonometric functions trigonometry maths

$\cot^{2} \dfrac{\pi}{11}+\cot^{2} \dfrac{2\pi}{11}+\cot^{2} \dfrac{3\pi}{11}........+\cot^{2} \dfrac{5\pi}{11}=?$

  1. $15$
  2. $45$
  3. $9$
  4. $18$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of cot^2(k*pi/11) for k=1 to 5 is a known trigonometric series result equal to 15.

Multiple choice trigonometric equations trigonometric functions trigonometry maths

$\tan \alpha  + 2\tan 2\alpha  + 4\tan 4\alpha  + 8\tan 8\alpha  + 16\tan 16\alpha  + 32\cot 32\alpha $ is equal

  1. $\cot \alpha $
  2. $\tan \alpha $
  3. $\cos \alpha $
  4. $sin \alpha $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a telescoping series of trigonometric functions. Using the identity tan(x) = cot(x) - 2cot(2x), the sum collapses to cot(alpha).

Multiple choice trigonometric equations trigonometric functions trigonometry maths

Simplify: $\tan5\tan { 30 } \times 4\tan { 85=\ _ \ _ \ _  } $

  1. $1$
  2. $4$
  3. $4/\surd 3$
  4. $4\surd 3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$\tan{{5}^{\circ}}\tan{{30}^{\circ}}\times 4\tan{{85}^{\circ}}$
$=4\tan{{5}^{\circ}}\times\dfrac{1}{\sqrt{3}}\tan{\left({90}^{\circ}-{5}^{\circ}\right)}$
$=4\tan{{5}^{\circ}}\times\dfrac{1}{\sqrt{3}}\cot{{5}^{\circ}}$
$=\dfrac{4}{\sqrt{3}}$ since $\tan{{5}^{\circ}}\cot{{5}^{\circ}}=1$