Get the solution to "A trigonometry exercise 2sin⁡π14+2sin⁡5π14−2sin⁡3π14=1"

trefoniu1

Answered question

2022-01-25

A trigonometry exercise

2 \sin \frac{π}{14} + 2 \sin \frac{5 π}{14} - 2 \sin \frac{3 π}{14} = 1

ul2ph3ojc

Beginner2022-01-26Added 12 answers

Using the formula
we need

S = 2 \cos \frac{3 π}{7} + 2 \cos \frac{π}{7} - 2 \cos \frac{2 π}{7}

\frac{2 π}{7} + \frac{5 π}{7} = π, \cos (π - x) = - \cos x

S = 2 \sum_{r = 0}^{2} \cos (2 r + 1) \frac{π}{7}

Anabelle Miller

Beginner2022-01-27Added 12 answers

Use the shorthand

a = \frac{π}{7}

2 \sin \frac{π}{14} + 2 \sin \frac{5 π}{14} - 2 \sin \frac{3 π}{14} = 2 \cos a + 2 \cos 3 a + 2 \cos 5 a

= \frac{1}{\sin a} (2 \sin a \cos a + 2 \sin a \cos 3 a + 2 \sin a \cos 5 a)

= \frac{1}{\sin a} (- \sin 0 a + \sin 2 a - \sin 2 a + \sin 4 a - \sin 4 a + \sin 6 a)

= \frac{\sin 6 a}{\sin a} = 1

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