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Math Help - trigonometry

  1. #1
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    trigonometry

    Find the values of x , valid between -pi and pi . I am confused because of the range .

    sin x =cos x

    My attempt .

    tan x =1

    so x will be 45 .. yes this is when x is between 0 and pi but how bout -pi and 0 ?? thanks
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by thereddevils View Post
    Find the values of x , valid between -pi and pi . I am confused because of the range .

    sin x =cos x

    My attempt .

    tan x =1

    so x will be 45 .. yes this is when x is between 0 and pi but how bout -pi and 0 ?? thanks
    you should take the angle when sin equal cos it is 45
    and 45-180=-135 in the third quarter since sin and cos are negative
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  3. #3
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    Quote Originally Posted by Amer View Post
    you should take the angle when sin equal cos it is 45
    and 45-180=-135 in the third quarter since sin and cos are negative

    Thanks Amer , i am not sure if what i did is correct .

    tan x =1

    x=45

    tan (-x) = 1

    -x=135 ( since quadrant 3 is positive )

    x=135

    Can i do like this ? Is my thought process correct >
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  4. #4
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    Quote Originally Posted by thereddevils View Post
    Find the values of x , valid between -pi and pi . I am confused because of the range .

    sin x =cos x

    My attempt .

    tan x =1

    so x will be 45 .. yes this is when x is between 0 and pi but how bout -pi and 0 ?? thanks
    Since the domain is given in radians the answer should also be given in radians (not degrees).

    x = \frac{\pi}{4} + n \pi where n is an integer. Now substitute different values of n to find all of the solutions that lie in the solution domain.
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  5. #5
    Senior Member pacman's Avatar
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    sin x = cos x,

    sin x/cos x = 1,

    tan x = 1,

    x = pi/4 + pi (k), where k is an integer, for domain -pi to pi,

    let k = -1 and 0

    (1) k = 0,

    x = pi/4,

    (2) k = -1,

    x = pi/4 - pi = -(3/4)pi

    see graph, ist quadrant sin x and cos x are positive, in the 3rd quadrant, both are negative.

    Attached Thumbnails Attached Thumbnails trigonometry-pi.gif  
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  6. #6
    Senior Member MacstersUndead's Avatar
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    If you need a different way to show why
    tan(x) = 1, draw the triangle with angle x with the opposite and adjacent sides equal to 1 [in the first quadrant]

    The hypotenuse is then sqrt(2), and you should recognize that the triangle is a special traingle most likely focused on in your class, where the angle of x is pi/4 radians (the reference angle)
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  7. #7
    Math Engineering Student
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    \sin x=\cos x\implies \sqrt{2}\sin \left( x-\frac{\pi }{4} \right)=0\implies x-\frac{\pi }{4}=k\pi ,\,k\in \mathbb{Z}.
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