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Question:

lostgirl
lostgirl
Student
High School - 12th Grade

Arctg(x) + Arcctg(1/3) = Pi/2.

Find the x value.

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Posted by lostgirl on Saturday October 3, 2009 at 5:31 AM and tagged with math, trigonometric identities, trigonometry.

Answers:

  1. neela
    neela Teacher
    Graduate School

    eNotes Editor

    arctgx+arctg (1/3)=pi/2

    To solve for x, we proceed as below:

    Taking tangent of the angles on both sides we get:

    tan(arctanx+arctan1/3)=tanpi/2

    [x+1/3]/(1-x*(1/3)]= +or-inf

    Threfore, denominator =0.

    Or 1-x/3=0 or x=3.

    The solution can also be got from a right angled triangle, ABC in which B is right angle:

    Given arctgx+arctg(1/3)=Pi/2

    In a right angled triangle with B =Pi/2, angle A+ angle C=Pi/2

    Let tan C = 1/3 = BC/AB, then , arctg (1/3 )= C by definition.

    Therefore Arctg x = arc tan A =arctg(AB/BC)= arctg(3)  or x= 3

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    Posted by neela on Saturday October 3, 2009 at 7:18 AM

  2. kjcdb8er
    kjcdb8er Teacher

    eNotes Editor

    You can answer this question by recalling (or looking up) an obscure trigonometric identity (see the link):

    tan(x) + atan(1/x) = pi/2 if x > 0

    .                             = -pi/2 if x < 0

    From the second term (1/x = 1/3), we know that x = 3

     

     

    To prove this identity, we can make use of well known pi/2 phase shift identities of trig functions, in this case tan(y) = cot(pi/2 - y)

    Let y = atan(x);   then x = tan(y) = cot(pi/2 - y)

    x = cot(pi/2 - y)

    acot(x) = acot(cot(pi/2 - y)) = pi/2 - y = pi/2 - atan(x)

    --> pi/2 = atan(x) + acot(x)

    This form is more well known than the first form given above. To get to that form, you have to recall that,

    acot(x) = atan(1/x) , x > 0

    acot(x) = -pi + atan(1/x) , x < 0

    Sources:
    http://en.wikipedia.org/wiki/List_of_trigonometric_identi...

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    Posted by kjcdb8er on Saturday October 3, 2009 at 8:04 AM

  3. lostgirl
    lostgirl Student
    High School - 12th Grade

    It`s arcctg (1/3) .. and as i know arcctg (x)=y and ctg (y)=x .

    So if arcctg(1/3)=x =>ctg (x)=1/3 . And i don`t know any value of x to solve this equation.

     

    It can be the exercise wrong?

     

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    Posted by lostgirl on Saturday October 3, 2009 at 1:28 PM

  4. kjcdb8er
    kjcdb8er Teacher

    eNotes Editor

    The identity is:

    atan(x) + acot(x) = pi/2

    Your equation is

    atan(x) + acot(1/3) = pi/2

    The only answer is x = 1/3

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    Posted by kjcdb8er on Sunday October 4, 2009 at 8:34 AM