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Sign of Trigonometric Ratios in Different Quadrants

Prove; cos ...

Question

Prove; $c o s \frac{π}{11} + c o s \frac{3 π}{11} + c o s \frac{5 π}{11} + c o s \frac{7 π}{11} + c o s \frac{9 π}{11} = \frac{1}{2}$

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Solution

Here the angles are in A.P. of 5 terms.
n = 5, $α = \frac{π}{11}, β = \frac{2 π}{1} ∴ \frac{β}{2} = \frac{π}{11}$
$S = \frac{s i n 5. \frac{π}{11}}{s i n \frac{π}{11}} . c o s \frac{1}{2} [\frac{π}{11} + \frac{9 π}{11}]$
$= \frac{2 s i n \frac{5 π}{11} c o s \frac{5 π}{11}}{2 s i n \frac{π}{11}} = \frac{s i n \frac{10 π}{11}}{2 s i n \frac{π}{11}} = \frac{s i n (π - \frac{π}{11})}{2 s i n \frac{π}{11}} = \frac{1}{2} .$

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Similar questions

Q. Find the value of

cos \frac{π}{11} + cos \frac{3 π}{11} + cos \frac{5 π}{11} + cos \frac{7 π}{11} + cos \frac{9 π}{11}

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Q. The value of

cos \frac{π}{11} + cos \frac{3 π}{11} + cos \frac{5 π}{11} + cos \frac{7 π}{11} + cos \frac{9 π}{11}

Q. Using eular's formula prove that

\frac{c o s π}{11} + \frac{c o s 3 π}{11} + \frac{c o 5 π}{11} + \frac{c o s 7 π}{11} + \frac{c o s 9 π}{11} = \frac{1}{2} s i n x

(1 + cos \frac{π}{8}) (1 + cos \frac{3 π}{8}) (1 + cos \frac{5 π}{8}) (1 + cos \frac{7 π}{8}) = \frac{1}{8}

(1 + cos \frac{π}{8}) (1 + cos \frac{3 π}{8}) (1 + cos \frac{5 π}{8}) (1 + cos \frac{7 π}{8}) = \frac{1}{8} .

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