Mathematics · Quantitative Aptitude

Trigonometric Identities and Values

74 Questions

Trigonometric identities and values questions require evaluating complex angles using standard formulas. These are critical for quantitative aptitude sections in SSC and various state exams. Success depends on memorizing standard values and applying transformation rules.

angle transformationstrigonometric ratiosstandard angle valuessine cosine products

Trigonometric Identities and Values Questions

Multiple choice using trigonometric tables trigonometric ratios of some specific angles trigonometric identities trigonometry maths

$\displaystyle3\tan^2{30^\circ}+\frac{4}{3}\cos^2{30^\circ}-2\sin^2{45^\circ}-\frac{1}{3}\sin^2{60^\circ}$ is equal to__________________.

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

Given 


$3\tan ^230+\dfrac 43\cos ^230-2\sin ^245-\dfrac 13\sin ^260$

$=3\left(\dfrac 1{\sqrt 3}\right)^2+\dfrac 43\left(\dfrac {\sqrt 3}2 \right)^2-2\left(\dfrac 1{\sqrt 2}\right)^2-\dfrac 13\left(\dfrac {\sqrt 3}2\right)^2$

$=1+1-1-\dfrac 14$

$= \dfrac 34$

Multiple choice maths average arithmetic mean of ap introduction to averages means

The artimetic mean of $2 sin 2^o, 4 sin 4^o, 6sin6^o,...,180sin 180^o$ is equal to 

  1. $cosec1^o$
  2. $sec1^o$
  3. $cot1^o$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a complex trigonometric series sum. The arithmetic mean of the given series is indeed cosec(1 degree).

Multiple choice

What is the value of (\sin 30^\circ) according to Aryabhata?

  1. \(\frac{1}{2}\)
  2. \(\frac{\sqrt{3}}{2}\)
  3. \(\frac{1}{\sqrt{2}}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Aryabhata used a table of sines to calculate the values of (\sin \theta) for (\theta = 0^\circ, 1^\circ, 2^\circ, ..., 90^\circ). According to his table, (\sin 30^\circ) is equal to (\frac{1}{2}).

Multiple choice

What is the value of (\cos 30^\circ) according to Aryabhata?

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

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\cos \theta). According to his table, (\cos 30^\circ) is equal to (\frac{\sqrt{3}}{2}).

Multiple choice

What is the value of (\sin 45^\circ) according to Aryabhata?

  1. \(\frac{1}{2}\)
  2. \(\frac{\sqrt{3}}{2}\)
  3. \(\frac{1}{\sqrt{2}}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\sin \theta). According to his table, (\sin 45^\circ) is equal to (\frac{1}{\sqrt{2}}).

Multiple choice

What is the value of (\cos 45^\circ) according to Aryabhata?

  1. \(\frac{1}{2}\)
  2. \(\frac{\sqrt{3}}{2}\)
  3. \(\frac{1}{\sqrt{2}}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\cos \theta). According to his table, (\cos 45^\circ) is equal to (\frac{1}{\sqrt{2}}).

Multiple choice

What is the value of (\sin 60^\circ) according to Aryabhata?

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

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\sin \theta). According to his table, (\sin 60^\circ) is equal to (\frac{\sqrt{3}}{2}).

Multiple choice

What is the value of (\cos 60^\circ) according to Aryabhata?

  1. \(\frac{1}{2}\)
  2. \(\frac{\sqrt{3}}{2}\)
  3. \(\frac{1}{\sqrt{2}}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Aryabhata used the Pythagorean identity (\sin^2 \theta + \cos^2 \theta = 1) to calculate the values of (\cos \theta). According to his table, (\cos 60^\circ) is equal to (\frac{1}{2}).

Multiple choice

What is the value of (\tan 60^\circ) according to Aryabhata?

  1. \(\frac{1}{2}\)
  2. \(\frac{\sqrt{3}}{2}\)
  3. \(\frac{1}{\sqrt{2}}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

Aryabhata used the definition of (\tan \theta) as (\frac{\sin \theta}{\cos \theta}) to calculate the values of (\tan \theta). According to his table, (\tan 60^\circ) is equal to (\sqrt{3}).

Multiple choice

What is the value of (\sin 75^\circ) according to Aryabhata?

  1. \(\frac{1}{2}\)
  2. \(\frac{\sqrt{3}}{2}\)
  3. \(\frac{1}{\sqrt{2}}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

Aryabhata used the half-angle formula for (\sin \theta) to calculate the values of (\sin \theta) for angles greater than (45^\circ). According to his table, (\sin 75^\circ) is equal to (\frac{\sqrt{6 + \sqrt{3}}}{4}).

Multiple choice

What is the value of (\cos 75^\circ) according to Aryabhata?

  1. \(\frac{1}{2}\)
  2. \(\frac{\sqrt{3}}{2}\)
  3. \(\frac{1}{\sqrt{2}}\)
  4. \(\frac{\sqrt{2}}{2}\)
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

Aryabhata used the half-angle formula for (\cos \theta) to calculate the values of (\cos \theta) for angles greater than (45^\circ). According to his table, (\cos 75^\circ) is equal to (\frac{\sqrt{6 - \sqrt{3}}}{4}).

Multiple choice

What is the value of (\cos 90^\circ) according to Aryabhata?

  1. \(\frac{1}{2}\)
  2. \(\frac{\sqrt{3}}{2}\)
  3. \(\frac{1}{\sqrt{2}}\)
  4. \(0\)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Aryabhata defined (\cos 90^\circ) to be equal to (0). This is because (\cos \theta) is the ratio of the adjacent side to the hypotenuse, and in a right triangle with an angle of (90^\circ), the adjacent side is equal to (0).