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JEE Advanced

Exam: JEE Advanced
Subject: Mathematics
Chapter: Trigonometry

JEE Advanced · Mathematics · Trigonometry

Find the value of 32 90^ 17 180^ 17 360^ 17 720^ 17 . If the sum of...

Find the value of

32 \sin \frac{90^{\circ}}{17} \cos \frac{180^{\circ}}{17} \cos \frac{360^{\circ}}{17} \cos \frac{720^{\circ}}{17}

. If the sum of the roots of the equation

\cos 4 x+6=7 \cos 2 x

in the interval

[0,314]

is

k \pi, k \in R

. Find k.

Correct answer

02

Step-by-step solution

Let

\theta=\left(\frac{90}{17}\right)^{\circ} \Rightarrow 17 \theta=90^{\circ}

Now,

32 \sin \frac{90^{\circ}}{17} \cos \frac{180^{\circ}}{17} \cos \frac{360^{\circ}}{17} \cos \frac{720^{\circ}}{17}=32 \sin \theta \cos 2 \theta \cos 4 \theta \cos 8 \theta

\begin{aligned} & =\frac{32 \cos \theta \sin \theta \cos 2 \theta \cos 4 \theta \cos 8 \theta}{\cos \theta}=2 \times \frac{\sin 16 \theta}{\cos \theta} \\ & =\frac{2 \sin (17 \theta-\theta)}{\cos \theta}=\frac{2 \sin \left(90^{\circ}-\theta\right)}{\cos \theta}=2\end{aligned}

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