Tag: trigonometric ratios of some specific angles

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Questions Related to trigonometric ratios of some specific angles

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

Values of : $sin{ 10 }^{ 0 }sin{ 50 }^{ 0 }sin{ 60 }^{ 0 }sin{ 70 }^{ 0 }$ is

  1. $\cfrac { 3 }{ 16 } $
  2. $\cfrac { 5 }{ 16 } $
  3. $\cfrac { \sqrt { 3 } }{ 16 } $
  4. $\cfrac { \sqrt { 5 } }{ 16 } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

As we know that

$\sin A\sin(60^{\circ}-A)\sin (60^{\circ}+A)=\dfrac{1}{4}\sin 3 A$
Put $A=10^{\circ}$
So $\sin 10^{\circ}\sin 50^{\circ}\sin 70^{\circ}=\dfrac{1}{4}\sin 30^{\circ}=\dfrac{1}{8}$
So $\sin 10^{\circ}\sin 60^{\circ}\sin 50^{\circ}\sin 70^{\circ}=\sin 60^{\circ}(\sin 10^{\circ}\sin 50^{\circ}\sin 70^{\circ})=\dfrac{\sqrt{3}}{2}\times \dfrac{1}{8}=\dfrac{\sqrt{3}}{16}$

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.