The sum of the roots of the equation $$4cos3x$$−$$4cos2x$$−$$cos$$π$$+x$$−$$1=0$$ in the interval $$0$$,$$315$$ is pπ , where p is equal to

Question

The sum of the roots of the equation  $$4cos3x$$−$$4cos2x$$−$$cos$$π$$+x$$−$$1=0$$ in the interval $$0$$,$$315$$ is pπ , where p is equal to

Infinity Learn · Accepted Answer

We have $$4cos3x$$−$$4cos2x$$−$$cos$$π$$+x$$−$$1=0$$            ⇒       $$4cos3x$$−$$4cos2x+cosx$$−$$1=0$$            ⇒$$4cos2x+1cosx$$−$$1=0$$             ⇒$$cosx=1$$⇒$$x=2n$$π              We  have    $$100$$π<$$315$$<$$101$$π∴Required sum $$=$$ $$2$$π$$+4$$π$$+$$.....$$.100$$π                                   $$=21+2+$$...$$.50$$π                                    $$=2$$×$$50$$×$$512$$π$$=2550$$π

Trigonometric equations

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The sum of the roots of the equation $4 \cos^{3} x - 4 \cos^{2} x - \cos (π + x) - 1 = 0$ in the interval $[0, 315]$ is $p π$ , where p is equal to

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Trigonometric equations

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The sum of the roots of the equation 4cos3x−4cos2x−cosπ+x−1=0 in the interval 0,315 is pπ , where p is equal to

We have 4cos3x−4cos2x−cosπ+x−1=0 ⇒ 4cos3x−4cos2x+cosx−1=0 ⇒4cos2x+1cosx−1=0 ⇒cosx=1⇒x=2nπ We have 100π<315<101π∴Required sum = 2π+4π+......100π =21+2+....50π =2×50×512π=2550π

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The sum of the roots of the equation $4 \cos^{3} x - 4 \cos^{2} x - \cos (π + x) - 1 = 0$ in the interval $[0, 315]$ is $p π$ , where p is equal to