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Please I need help in these exercises: tan (x+π/4)=2tanx + 2  

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postpunk | eNotes Newbie

Posted July 10, 2013 at 1:46 PM via web

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Please I need help in these exercises:

tan (x+π/4)=2tanx + 2

 

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted July 10, 2013 at 2:14 PM (Answer #1)

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Solve `tan(x+pi/4)=2tanx+2` :

Now `tan(A+B)=(tanA+tanB)/(1-tanAtanB)` so we can rewrite the left-hand side:

`(tanx+tan(pi/4))/(1-tanxtan(pi/4))=2tanx+2` and `tan(pi/4)=1` so we have:

`(tanx+1)/(1-tanx)=2tanx+2` Multiply both sides by (1-tanx)

`tanx+1=2-2tan^2x`

`2tan^2x+tanx-1=0`

`(2tanx-1)(tanx+1)=0` By the zero product property

`tanx=1/2 "or" tanx=-1`

If tanx=-1 then `x=-pi/4+npi,n in ZZ` (n an integer)

If `tanx=1/2 ==> x=tan^(-1)1/2==>x~~.464+npi,n in ZZ`

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The solutions are `x=-pi/4+npi,x~~.464+npi`

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The graph of the left side in black, the right side in red:

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