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Math Help - Why is cos pi = -1?

  1. #1
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    Why is cos pi = -1?

    I do not know why cos pi is equivalent to -1. Generally, on the unit circle I've been learning of since a day ago, it shows all the fractions of pi arranged around the circle. Each pi fraction extends from the centre of the coordinate plane and forms a right angle triangle. With most of the right triangles I have a sort of intuitive grasp of them. But I don't see any trangle that can be constructed on the unit circle that would make sense of cos pi = -1.

    I mean, with any triangle wouldn't the hypotunse being some value like the root of 3 prevent cos pi from being -1? I mean, -1 over 3 pi or something would make sense. But just -1?

    I'm sorry about not be able to provide a more detailed attempted answer. I'm also sorry for the inelegance of this post. I just really do not have a good grasp of the unit circle. I know my thoughts are probably quite far off the correct answer.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by D. Martin View Post
    I do not know why cos pi is equivalent to -1. Generally, on the unit circle I've been learning of since a day ago, it shows all the fractions of pi arranged around the circle. Each pi fraction extends from the centre of the coordinate plane and forms a right angle triangle. With most of the right triangles I have a sort of intuitive grasp of them. But I don't see any trangle that can be constructed on the unit circle that would make sense of cos pi = -1.

    I mean, with any triangle wouldn't the hypotunse being some value like the root of 3 prevent cos pi from being -1? I mean, -1 over 3 pi or something would make sense. But just -1?

    I'm sorry about not be able to provide a more detailed attempted answer. I'm also sorry for the inelegance of this post. I just really do not have a good grasp of the unit circle. I know my thoughts are probably quite far off the correct answer.
    don't think of it in terms of triangles. think in terms of coordinates.

    recall that sine and cosine can be defined in terms of the points on the units circle.

    the points on the unit circle can be parameterized by x = \cos \theta and y = \sin \theta, where \theta is the angle formed by the x-axis and the line connecting the origin to the point (x,y) = (\cos \theta , \sin \theta) on the circle.

    when we are at \pi ( 180^\circ), the points on the circle is (-1,0), so that (\cos \pi, \sin \pi) = (-1, 0), and you just equate the coordinates from there
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  3. #3
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    Quote Originally Posted by D. Martin View Post
    I do not know why cos pi is equivalent to -1. Generally, on the unit circle I've been learning of since a day ago, it shows all the fractions of pi arranged around the circle. Each pi fraction extends from the centre of the coordinate plane and forms a right angle triangle. With most of the right triangles I have a sort of intuitive grasp of them. But I don't see any trangle that can be constructed on the unit circle that would make sense of cos pi = -1.

    I mean, with any triangle wouldn't the hypotunse being some value like the root of 3 prevent cos pi from being -1? I mean, -1 over 3 pi or something would make sense. But just -1?

    I'm sorry about not be able to provide a more detailed attempted answer. I'm also sorry for the inelegance of this post. I just really do not have a good grasp of the unit circle. I know my thoughts are probably quite far off the correct answer.
    Here is one way of explaining why cos(pi) = -1. A way using your "triangle" in imagining angles.

    Say an angle is not flat. (Pi is a flat angle.) Say an angle is "opened", but not pi/2 nor 3pi/2. So we can draw the reference triangle of the angle in the unit circle.

    The opp ...opposite side... is the y-coordinate
    The adj ...adjacent side... is the x-coordinate
    The hyp is the hypotenuse

    The cosine function of the angle is adj/hyp......(adjacent side)/(hypotenuse)
    So, the cos is (x-coordinate)/(hypotenuse).

    Now we go to when the angle is flat, or pi.
    What is the x-coordinate? It is -1
    What is the hypotenuse? It is 1
    So, cos(pi) = (-1)/(1) = -1
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