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Math Help - Graph question :S

  1. #1
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    Exclamation Graph question :S

    The question is:
    9(i) Sketch, on a single diagram showing values of x from -180degs to +180degs, the graph of y=tanx and y= 4cos x.

    The equation tan x = 4cos x

    has two roots in the interval -180degs\< +\<180degs. These are denoted by a(alpha) and b(pheta), where a<b

    (ii) Show a and b on your sketch, and express b in terms of a.

    (iii) Show that the equation tan x = 4cos x may be written as 4sin^2 x + sin x - 4 =0

    Hence find the value of b-a, correct to the nearest degree.

    If you can help me in any way possible, i have drawn the graph!
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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by LoveDeathCab View Post
    (iii) Show that the equation tan x = 4cos x may be written as 4sin^2 x + sin x - 4 =0
    This has already been answered in a previous post.
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  3. #3
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    I know, its the rest that i am struggling with.
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  4. #4
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    Hello, LoveDeathCab!

    9(i) Sketch, on one diagram, the graphs: .y = tan(x) and y = 4Ěcos(x), -180░ < x < +180░

    The equation: tan(x) = 4Ěcos(x) has two roots in the interval [-180░, +180░]
    These are denoted by α and β, where α < β.
    We see that the graphs intersect on [0░, 90░] and [90░, 180░]


    (ii) Show α and β on your sketch, and express β in terms of α.
    Are they kidding? . . . We need part (iii) first, don't we? .**


    (iii) Show that the equation: tan(x) = 4Ěcos(x) may be written as: 4Ěsin▓(x) + sin(x) - 4 .= .0

    Hence find the value of β - α, correct to the nearest degree.
    So that's where that equation came from!


    We have a quadratic: .4Ěsin▓(x) + sin(x) - 4 .= .0
    . . . . . . . . . . . . . . . . . . . . . . . . . . __
    . . . . . . . . . . . . . . . . . . . . . .-1 ▒ √65
    Quadratic Formula: . sin(x) .= .-----------
    . . . . . . . . . . . . . . . . . . . . . . . . 8

    . . . . . . . . . . . . . . . . . . . __
    We have: .sin(x) .= .(-1 - √65)/8 .= .-1.132782219 ... no real roots

    . . . . . . . . . . . . . . . . . . . . . . .__
    And we have: .sin(x) .= .(-1 + √65)/8 .= .0.882782219
    . . Hence: .x . .62░, 118░


    Therefore: .β - α .= .118░ - 62░ .= .56░

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    **

    Even if we did part (iii) first, it's still pretty awful . . .

    The equation: .sin▓(x) + ╝Ěsin(x) - 1 .= .0 has two roots: α and β.

    . . Then: .sin(α) + sin(β) .= .-╝

    Solve for β: .β .= .arcsin(-sinα - ╝)

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