Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20199 AprilEvening Shift
Exam
JEE Main
Year
2019
Date
9 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

JEE Main 2019 · 9 April · Evening Shift · Mathematics · Trigonometric Ratios & Identities

The value of 10 ° 30 ° 50 ° 70 ° is:

The value of sin⁡10°sin⁡30°sin⁡50°sin⁡70°\sin 10 ° \sin 30 ° \sin 50 ° \sin 70 ° is:

Options

  1. A136\dfrac{1}{36}
  2. B116\dfrac{1}{16}
  3. C118\dfrac{1}{18}
  4. D132\dfrac{1}{32}

Correct answer

B. 116\dfrac{1}{16}

Step-by-step solution

sin⁡10°sin⁡30°sin⁡50°sin⁡70°\sin 10 ° \sin 30 ° \sin 50 ° \sin 70 °

=sin⁡10°×12(12×2sin⁡50°sin⁡70°)= \sin 10 ° \times \dfrac{1}{2}\left(\dfrac{1}{2} \times 2\sin 50 ° \sin 70 °\right)

=14sin⁡10°(2sin⁡50°sin⁡70°)= \dfrac{1}{4}\sin 10 ° \left(2\sin 50 ° \sin 70 °\right)

Using 2sinAsinB=cos⁡(A−B)−cos⁡(A+B)2sinAsinB = \cos \left(A - B\right) - \cos \left(A + B\right)

=14sin⁡10°(cos⁡20°−cos⁡120°)= \dfrac{1}{4}\sin 10 ° \left(\cos 20 ° - \cos 120 °\right)

=14sin⁡10°(cos⁡20°−cos⁡(90°+30°))= \dfrac{1}{4}\sin 10 ° \left(\cos 20 ° - \cos \left(90 ° + 30 °\right)\right)

=14sin⁡10°(cos⁡20°+sin⁡30°)= \dfrac{1}{4}\sin 10 ° \left(\cos 20 ° + \sin 30 °\right)

=14sin⁡10°(cos⁡20°+12)= \dfrac{1}{4}\sin 10 ° \left(\cos 20 ° + \dfrac{1}{2}\right)

=18(2sin⁡10°cos⁡20°+sin⁡10°)= \dfrac{1}{8}\left(2\sin 10 ° \cos 20 ° + \sin 10 °\right)

Again, using 2sinAcosB=sin⁡(A+B)−sin⁡(A−B)2sinAcosB = \sin \left(A + B\right) - \sin \left(A - B\right)

=18(sin⁡30°−sin⁡10°+sin⁡10°)= \dfrac{1}{8}\left(\sin 30 ° - \sin 10 ° + \sin 10 °\right)

=18(12)=116.= \dfrac{1}{8}\left(\dfrac{1}{2}\right) = \dfrac{1}{16} .

Practice Trigonometric Ratios & Identities on Quantrex Academy →

More from Trigonometric Ratios & Identities

Let α = (1 - 2 ((π)/(11)))(1 - 2 ((3π)/(11)))(1 - 2 ((9π)/(11)))(1 - 2 ((27π)/(11)))(1… JEE Advanced 2026If A = ( 3^∘)/( 9^∘) + ( 9^∘)/( 27^∘) + ( 27^∘)/( 81^∘) and B = 81^∘ - 3^∘ , then (B)/(A)… JEE Main 2026Let P = \ θ ∈ [0, 4π] : ^2θ ≠ 1\ and S = \ a ∈ ℤ : 2( ^8θ - ^8θ) 2θ = a^2, θ ∈ P\ . Then… JEE Main 2026If ((π)/(18)) ((5π)/(18)) ((7π)/(18)) = K , then the value of ((10Kπ)/(3)) is : JEE Main 2026If (A-B) A + ^ 2 C ^ 2 ~A =1, ~A, ~B, C ∈(0, (π)/(2)) , then JEE Main 2026If x=(5)/(12) for some x ∈(π, (3 π)/(2)) , then 7 x( (13 x)/(2)+ (13 x)/(2))+ 7 x( (13… JEE Main 2026The value of 3 cosec 20^ ∘ - 20^ ∘ 20^ ∘ 40^ ∘ 60^ ∘ 80^ ∘ is equal to JEE Main 2026The least value of ( ^ 2 θ-6 θ θ+3 ^ 2 θ+2) is JEE Main 2026Let (π)/(2) ((15 θ)/(2))( 8 θ+ 8 θ)+ ((15 θ)/(2))( 8 θ- 8 θ) is equal to JEE Main 2026Let α and β respectively be the maximum and the minimum values of the function f(θ)=4( ^… JEE Main 2026Let (α+β)=-(1)/(10) and (α-β)=(3)/(8) , where 0<α<(π)/(3) and 0<β<(π)/(4) . If 2 α= 3(1-r… JEE Main 2026If ^ 2 48^ ∘ - ^ 2 12^ ∘ ^ 2 24^ ∘ - ^ 2 6^ ∘ = α+β 5 2 , where α, β ∈ ℕ , then α+β is… JEE Main 2026The value of cosec 10^ ∘ - 3 10^ ∘ is equal to : JEE Main 2026Let α= 1 60^ ∘ 61^ ∘ + 1 62^ ∘ 63^ ∘ +…+ 1 118^ ∘ 119^ ∘ Then the value of ( cosec 1^ ∘ α… JEE Advanced 2025If x+ ^2 x=1, x ∈(0, (π)/(2)) , then ( ^ 12 x+ ^ 12 x)+3( ^ 10 x+ ^ 10 x+ ^8 x+ ^8 x)+(… JEE Main 2025If ∑_ r=1 ^ 13 \ (1)/( ((π)/(4)+(r-1) (π)/(6)) ((π)/(4)+(r π)/(6)))\ =a 3 +b, a, b ∈ Z… JEE Main 2025 Full Trigonometric Ratios & Identities list Trigonometric Ratios & Identities JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs