Mathematics · Quantitative Aptitude

Trigonometric Identities and Values

70 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

$sin^21^0+sin^22^0+sin^23^0+....+sin^290^0$

  1. 0

  2. 1

  3. 89/2

  4. 91/2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using the identity sin^2(90 - theta) = cos^2(theta), the sum can be paired from both ends. There are 89 terms from 1 to 89 degrees plus sin^2(90) = 1. Pairing sin^2(theta) + cos^2(theta) gives 44 pairs equal to 1, plus sin^2(45) = 1/2, plus sin^2(90) = 1, totaling 44 + 1/2 + 1 = 91/2.

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

$\frac { cos{ 70 }^{ \circ  } }{ sin{ 20 }^{ \circ  } } +\frac { cos{ 59 }^{ \circ  } }{ sin{ 31 }^{ \circ  } } -8{ sin }^{ 2 }{ 30 }^{ \circ  }$

  1. $1$
  2. $-1$
  3. $0$
  4. $2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since cos(70 degrees) = sin(20 degrees) and cos(59 degrees) = sin(31 degrees), the first two fractions each equal 1. The term 8 * sin^2(30 degrees) = 8 * (1/2)^2 = 8 * (1/4) = 2. Thus, 1 + 1 - 2 = 0.

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

The value of $(4 \, cos^2 9^o - 1) (4 cos^2 27^o - 1) (4 \, cos^2 81^o - 1) (4 \, cos^2 243^o - 1) $ is 

  1. 1

  2. -1

  3. 2

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

This is a product of the form (4cos^2(x)-1). Using the identity 4cos^2(x)-1 = sin(3x)/sin(x), the product telescopes to sin(3^n * x) / (sin(x) * 2^n). For the given angles, the result is 1.

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

$\sqrt {3}\csc 20^{o}-\sec 20^{o}$ is equal to

  1. $2$
  2. $2\sin 20^{o}\csc 40^{o}$
  3. $4$
  4. $4\sin 20^{o}\csc 40^{o}$.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Rewrite the expression as sqrt(3)/sin(20) - 1/cos(20) = (sqrt(3)cos(20) - sin(20)) / (sin(20)cos(20)). Multiplying numerator and denominator by 2 gives (2 * (sqrt(3)/2 cos(20) - 1/2 sin(20))) / (1/2 * 2 sin(20)cos(20)) = 2 * sin(60 - 20) / (1/2 sin(40)) = 2 * sin(40) / (1/2 sin(40)) = 4.

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

Find the value of, $\dfrac {4}{3}\cot^{2}30^{o}+\cot^{2}60^{o}-2\csc ^{2}60^{o}-\dfrac {3}{4}\tan^{2}30^{o}$

  1. $10/3$
  2. $11/3$
  3. $4$
  4. $none\ of\ these$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\dfrac { 4 }{ 3 } { \cot }^{ 2 }30+{ \cot }^{ 2 }60-2{ csc }^{ 2 }60-\dfrac { 3 }{ 4 } { \tan }^{ 2 }30$

$\Rightarrow \dfrac { 4 }{ 3 } { \left( \sqrt { 3 }  \right)  }^{ 2 }+{ \left( \dfrac { 1 }{ \sqrt { 3 }  }  \right)  }^{ 2 }-2\times { \left( \dfrac { 2 }{ \sqrt { 3 }  }  \right)  }^{ 2 }-\dfrac { 3 }{ 4 } \times { \left( \dfrac { 1 }{ \sqrt { 3 }  }  \right)  }^{ 2 }$
$\Rightarrow 4+\dfrac { 1 }{ 3 } -\dfrac { 8 }{ 3 } -\dfrac { 1 }{ 4 } $
$\Rightarrow \dfrac { 15 }{ 4 } -\dfrac { 7 }{ 3 } $
$=\dfrac { 17 }{ 12 } $
None of these.

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

$ \sqrt { 3 } \ cosec 20 ^ { \circ } - \sec 20 ^ { \circ }$  is equal to :

  1. $2$
  2. $2 \sin 20 ^ { \circ } / \sin 40 ^ { \circ }$
  3. $4$
  4. $4 \sin 20 ^ { \circ } / \sin 40 ^ { \circ }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Expressing csc(20) as 1/sin(20) and sec(20) as 1/cos(20), the expression becomes sqrt(3)/sin(20) - 1/cos(20). Combining terms and using the sine subtraction formula with a factor of 2 yields 4, identical to the standard identity reduction.

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

If $\alpha =685^o$, then $(\cos\alpha -\sin\alpha)$ is equivalent to?

  1. $-\cos 35^o-\sin 35^o$
  2. $-\cos 35^o +\sin 35^o$
  3. $\cos 35^o-\sin 35^o$
  4. $\cos 35^o+\sin 35^o$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Reduce alpha = 685 degrees modulo 360 degrees: 685 - 2*360 = 685 - 720 = -35 degrees. Then cos(-35 degrees) - sin(-35 degrees) = cos(35 degrees) + sin(35 degrees) since cosine is even and sine is odd.

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).