If $$16sin$$θ,$$cos$$θ and $$tan$$θ are in GP, and range of θ is $$0$$,nπ and has $$5$$ values, then n $$=$$

Question

If  $$16sin$$θ,$$cos$$θ and  $$tan$$θ  are in GP, and range of θ is $$0$$,nπ and  has $$5$$ values, then n $$=$$

Infinity Learn · Accepted Answer

Since, $$16sin$$θ,$$cos$$θ,$$tan$$θ are in G.P.∴$$cos2$$θ$$=16sin$$θ.$$tan$$θ$$cos2$$θ $$=16sin2$$θ$$cos$$θ  put $$sin2$$θ $$=1-cos2$$θ⇒$$6cos3$$θ$$+cos2$$θ−$$1=0Note$$ that  $$cos$$θ$$=12$$ satisfies the equation (by$$ $$trial)∴(2cos$$θ−$$1)(3cos2$$θ$$+2cos$$θ$$+1)=0$$⇒$$cos$$θ$$=12$$           $$(other$$ $$cos$$θ values of  are $$imaginary)⇒$$cos$$θ$$=$$$$cos$$$$π$$$$3$$⇒θ$$=2n$$π±π$$3$$,n∈Z⇒for $$n=0$$, θ$$=$$±π$$3$$     ∴θ$$=$$π$$3$$,$$2$$π$$-$$π$$3$$,$$2$$π$$+$$π$$3$$,$$4$$π$$-$$π$$3$$ and $$4$$π$$+$$π$$3are$$ the $$5$$ values taken by θ in $$0$$,$$5$$π ⇒$$n=5$$   for  $$n=1$$, θ$$=2$$π±π$$3$$     for   $$n=2$$, θ$$=4$$π±π$$3$$

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If $\frac{1}{6} \sin θ, \cos θ$ and $\tan θ$ are in GP, and range of $θ$ is $[0, n π]$ and has 5 values, then n =

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Trigonometric equations

Remember concepts with our Masterclasses.

If 16sinθ,cosθ and tanθ are in GP, and range of θ is 0,nπ and has 5 values, then n =

Ready to Test Your Skills?

free study material

If $\frac{1}{6} \sin θ, \cos θ$ and $\tan θ$ are in GP, and range of $θ$ is $[0, n π]$ and has 5 values, then n =