The number of values of θ in [$$0$$,$$3$$π] satisfying $$sin3$$θ$$+cos3$$θ$$+3sin$$θ$$cos$$θ$$-1=0$$ is

Question

The number of values of θ in  [$$0$$,$$3$$π] satisfying $$sin3$$θ$$+cos3$$θ$$+3sin$$θ$$cos$$θ$$-1=0$$ is

Infinity Learn · Accepted Answer

We have  $$sin3$$θ$$+cos3$$θ−$$1=$$−$$3sin$$θ$$cos$$θ⇒$$sin3$$θ$$+cos3$$θ$$+(-1)3=3($$$$sin$$θ$$)($$$$cos$$θ$$)($$−$$1)which$$ is in the form       $$a3+b3+c3=3abc$$⇒∴$$sin$$θ$$=$$$$cos$$θ$$=$$−$$1$$  which is impossibleor   $$sin$$θ$$+$$$$cos$$θ$$+($$−$$1)=0$$⇒$$sin$$θ$$+$$$$cos$$θ$$=1$$⇒$$12sin$$θ$$+12cos$$θ$$=12$$⇒$$cos$$θ−π$$4=$$$$cos$$$$π$$$$4$$⇒θ−π$$4=2n$$π±π$$4$$⇒θ$$=2n$$π$$+$$π$$2$$     or     θ$$=$$  $$2n$$π⇒ for $$n=0$$,θ$$=$$π$$2$$,$$0$$      and    for   $$n=1$$,    θ$$=5$$$$π$$$$2$$,$$2$$π

The number of values of $θ$ in $[0, 3 π]$ satisfying $\sin^{3} θ + \cos^{3} θ + 3 \sin θ \cos θ - 1 = 0$ is

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The number of values of θ in [0,3π] satisfying sin3θ+cos3θ+3sinθcosθ-1=0 is

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The number of values of $θ$ in $[0, 3 π]$ satisfying $\sin^{3} θ + \cos^{3} θ + 3 \sin θ \cos θ - 1 = 0$ is