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Math Help - trig proof

  1. #1
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    trig proof

    Use what you learned in calculus to show that Sin(alpha)≤alpha and sin(alpha)/cos(alpha)≥alpha holds for all 0≤alpha≤pi/2
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by rmpatel5 View Post
    Use what you learned in calculus to show that Sin(alpha)≤alpha and sin(alpha)/cos(alpha)≥alpha holds for all 0≤alpha≤pi/2
    sin(alpha)≤alpha <--> sin(alpha)-alpha≤0
    Take the derivative of sin(alpha)-alpha and see its variations between 0 and pi/2

    Same goes for the other one.
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  3. #3
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    Quote Originally Posted by Moo View Post
    Hello,

    sin(alpha)≤alpha <--> sin(alpha)-alpha≤0
    Take the derivative of sin(alpha)-alpha and see its variations between 0 and pi/2

    Same goes for the other one.
    I took the der. of the first one and i get cos(alpha)≤1 and the second one sec^2(x)≥1

    the first case:
    alpha goes from 0 to pi/2

    second case:
    is just alpha is 0

    i dont think my answer makes sense
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  4. #4
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    You can use it using geometry.
    Look at the picture.
    The angle between the x-axis and the red line is \alpha.
    While the circle is a unit circle.
    Thus, the length of the arc made by \alpha is \alpha.
    While the length of the blue line is \sin \alpha.

    Now reflect that image through the x-axis (second picture).
    We see that the length of the blue line is 2\sin \alpha.
    And the length of arc made by the blue line is 2\alpha.
    Thus, 2\sin \alpha \leq 2\alpha \implies \sin \alpha \leq \alpha.
    This is because the shortest distance between two points is a straight line.
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