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Why is $x$ in Figure 1 $x$ and not $-x$? This has caused me to not understand why $\cos(-\theta) = x$ and $\cos(\theta) = x$.

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That's just sloppy drafting of the figure. For the angles shown, the cosine is indeed negative. –  Henning Makholm Apr 25 '12 at 1:12
    
because $x$ is the projection of the point $(x,y)$ over the X-axis, wich represent $\cos\theta$ –  Abdelmajid Khadari Apr 25 '12 at 1:12

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The point has co-ordinates $(x,y)$. In the diagram, $x$ is a negative number, but that's OK - variables are allowed to stand for negative numbers. The angle $\theta$ is in the second quadrant, where the cosine function is negative, so $\cos\theta=x$ is perfectly consistent.

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I understood the OP's problem as being that the diagram seems to claim that the length of the line segment between the origin and $(\cos\theta,0)$ is $x$ -- but actually it is $|x|$, which in this case equals $-x$. –  Henning Makholm Apr 25 '12 at 10:30
    
@Henning, you might be right. –  Gerry Myerson Apr 25 '12 at 10:58

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