The least positive integer value of n > 3 satisfying the equation 1 sin π n = 1 sin 2 π n + 1 sin 3 π n is

Given equation is 1 sin π n = 1 sin 2 π n + 1 sin 3 π n . ⇒ 1 sin π n - 1 sin 3 π n = 1 sin 2 π n ⇒ sin 3 π n - sin π n sin π n sin 3 π n = 1 sin 2 π n ⇒ 2 cos 3 π n + π n 2 sin 3 π n - π n 2 sin π n sin 3 π n = 1 sin 2 π n ∵ sin C - sin D = 2 sin C - D 2 cos C + D 2 ⇒ 2 cos 2 π n sin π n sin π n sin 3 π n = 1 sin 2 π n ⇒ 2 cos 2 π n sin 3 π n = 1 sin 2 π n ⇒ 2 cos 2 π n sin 2 π n = sin 3 π n ∵ sin 2 A = 2 sin A cos A ⇒ sin 4 π n = sin 3 π n 4 π n = m π + - 1 m 3 π n , m ∈ I ∵ sin α = sin β , α = r π + - 1 n β , r ∈ I Case 1 : If m = 2 λ 4 π n = 2 λ π + - 1 2 λ 3 π n So, - 1 2 λ will always positive ⇒ π n = 2 λ π n = 1 2 λ , which is not possible. Case 2 : If m = 2 λ + 1 4 π n = 2 λ + 1 π + - 1 2 λ + 1 3 π n So, - 1 2 λ + 1 will always negative. ⇒ 7 π n = 2 λ + 1 π ⇒ 7 n = 2 λ + 1 For λ = 0 , n = 7 .

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 12 - Trigonometric Equations and Inequalities>Exercise-2>Q 42

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Earn 100

$\cos 15 x = \sin 5 x$ if

50% studentsanswered this correctly

Important Questions on Trigonometric Equations and Inequalities

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Alpha Question Bank for Engineering: Mathematics>Chapter 12 - Trigonometric Equations and Inequalities>Exercise-2>Q 43

$5 \sin^{2} x + \sqrt{3} \sin x \cos x + 6 \cos^{2} x = 5$ if

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Alpha Question Bank for Engineering: Mathematics>Chapter 12 - Trigonometric Equations and Inequalities>Exercise-2>Q 44

$\sin^{2} x + 2 \sin x \cos x - 3 \cos^{2} x = 0$ if

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Alpha Question Bank for Engineering: Mathematics>Chapter 12 - Trigonometric Equations and Inequalities>Exercise-2>Q 45

Solution set of inequality

\sin^{3} x \cos x > \cos^{3} x \sin x,

where

x \in (0, π)

, is

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Alpha Question Bank for Engineering: Mathematics>Chapter 12 - Trigonometric Equations and Inequalities>Exercise-3>Q 1

\cos (α - β) = 1

and

\cos (α + β) = \frac{1}{e},

where

α, β \in [- π, π] .

Pairs

α, β

which satisfy both the equations is/are

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Alpha Question Bank for Engineering: Mathematics>Chapter 12 - Trigonometric Equations and Inequalities>Exercise-3>Q 2

Let

θ \in (0, \frac{π}{4})

and

t_{1} = {(\tan θ)}^{\tan θ}, t_{2} = {(\tan θ)}^{\cot θ}, t_{3} = {(\cot θ)}^{\tan θ}

and

t_{4} = {(\cot θ)}^{\cot θ},

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 12 - Trigonometric Equations and Inequalities>Exercise-3>Q 3

If

0 < θ < 2 π,

then the intervals of values of

θ

for which

2 \sin^{2} θ - 5 \sin θ + 2 > 0,

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 12 - Trigonometric Equations and Inequalities>Exercise-3>Q 4

The number of solutions of the pair of equations

2 \sin^{2} θ - \cos 2 θ = 0, 2 \cos^{2} θ - 3 sinθ = 0

in the interval

[0, 2 π]

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 12 - Trigonometric Equations and Inequalities>Exercise-3>Q 6

The least positive integer value of $n > 3$ satisfying the equation $\frac{1}{\sin (\frac{π}{n})} = \frac{1}{\sin (\frac{2 π}{n})} + \frac{1}{\sin (\frac{3 π}{n})}$ is