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sin-x-3cos-x-tan-x-3-True-or-false-




Question Number 12905 by 433 last updated on 06/May/17
sin (x)=3cos (x) ⇔tan (x)=3  True or false?
$$\mathrm{sin}\:\left({x}\right)=\mathrm{3cos}\:\left({x}\right)\:\Leftrightarrow\mathrm{tan}\:\left({x}\right)=\mathrm{3} \\ $$$${True}\:{or}\:{false}? \\ $$
Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 06/May/17
1)false if: cosx=0⇒x=2kπ+(π/2)  2)true if x≠2kπ+(π/2),k∈Z .
$$\left.\mathrm{1}\right){false}\:{if}:\:{cosx}=\mathrm{0}\Rightarrow{x}=\mathrm{2}{k}\pi+\frac{\pi}{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){true}\:{if}\:{x}\neq\mathrm{2}{k}\pi+\frac{\pi}{\mathrm{2}},{k}\in\boldsymbol{{Z}}\:. \\ $$
Commented by mrW1 last updated on 07/May/17
if cos x=0, then sin x=±1≠3×0  ⇒cos x≠0  therefore:  sin x=3cos x  ⇔tan x=3
$${if}\:\mathrm{cos}\:{x}=\mathrm{0},\:{then}\:\mathrm{sin}\:{x}=\pm\mathrm{1}\neq\mathrm{3}×\mathrm{0} \\ $$$$\Rightarrow\mathrm{cos}\:{x}\neq\mathrm{0} \\ $$$${therefore}: \\ $$$$\mathrm{sin}\:{x}=\mathrm{3cos}\:{x} \\ $$$$\Leftrightarrow\mathrm{tan}\:{x}=\mathrm{3} \\ $$
Answered by mrW1 last updated on 07/May/17
true!
$${true}! \\ $$
Commented by prakash jain last updated on 07/May/17
a)  sin x=3cos x⇒((sin x)/(cos x))=3⇒tan x=3  b)  tan x=3⇒((sin x)/(cos x))=3⇒sin x=3cos x  hence  sin x=3cos x⇔tan x=3
$$\left.{a}\right) \\ $$$$\mathrm{sin}\:{x}=\mathrm{3cos}\:{x}\Rightarrow\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}=\mathrm{3}\Rightarrow\mathrm{tan}\:{x}=\mathrm{3} \\ $$$$\left.{b}\right) \\ $$$$\mathrm{tan}\:{x}=\mathrm{3}\Rightarrow\frac{\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}=\mathrm{3}\Rightarrow\mathrm{sin}\:{x}=\mathrm{3cos}\:{x} \\ $$$${hence} \\ $$$$\mathrm{sin}\:{x}=\mathrm{3cos}\:{x}\Leftrightarrow\mathrm{tan}\:{x}=\mathrm{3} \\ $$
Commented by prakash jain last updated on 07/May/17
question requires ⇔ so both  needs to be shown.
$$\mathrm{question}\:\mathrm{requires}\:\Leftrightarrow\:\mathrm{so}\:\mathrm{both} \\ $$$$\mathrm{needs}\:\mathrm{to}\:\mathrm{be}\:\mathrm{shown}.\: \\ $$

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