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Question Number 126954 by bramlexs22 last updated on 25/Dec/20

$$\cos \left(\frac{5\pi }{18}\right)+\cos \left(\frac{7\pi }{18}\right)+\cos \left(\frac{17\pi }{18}\right)=?$$

Answered by Dwaipayan Shikari last updated on 25/Dec/20

$$2cos\left(\frac{12\pi }{36}\right)cos\left(\frac{\pi }{18}\right)+cos\left(\frac{17\pi }{18}\right)$$$$=cos\left(\frac{\pi }{18}\right)+cos\left(\frac{17\pi }{18}\right)=2cos\left(\frac{\pi }{2}\right)cos\left(\frac{8\pi }{18}\right)=0$$

Commented by bramlexs22 last updated on 25/Dec/20

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Answered by Olaf last updated on 25/Dec/20

$$\mathrm{X}=\cos \left(\frac{5\pi }{18}\right)+\cos \left(\frac{7\pi }{18}\right)+\cos \left(\frac{17\pi }{18}\right)$$$$\mathrm{X}=\cos (\frac{\pi }{3}-\frac{\pi }{18})+\cos (\frac{\pi }{3}+\frac{\pi }{18})+\cos (\pi -\frac{\pi }{18})$$$$\mathrm{X}=\cos \left(\frac{\pi }{3}\right)\cos \left(\frac{\pi }{18}\right)-\sin \left(\frac{\pi }{3}\right)\sin \left(\frac{\pi }{18}\right)$$$$+\cos \left(\frac{\pi }{3}\right)\cos \left(\frac{\pi }{18}\right)+\sin \left(\frac{\pi }{3}\right)\sin \left(\frac{\pi }{18}\right)-\cos \left(\frac{\pi }{18}\right)$$$$\mathrm{X}=2\cos \left(\frac{\pi }{3}\right)\cos \left(\frac{\pi }{18}\right)-\cos \left(\frac{\pi }{18}\right)$$$$\mathrm{X}=2.\frac{1}{2}.\cos \left(\frac{\pi }{18}\right)-\cos \left(\frac{\pi }{18}\right)=0$$

Commented by bramlexs22 last updated on 25/Dec/20

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