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What does the secant value represent?

I know that $$\sec = 1/\cos(\theta)$$ but really I do not know what this value represents, so I need your help. A clear example with images would be appreciated.

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@Rahul: Not actually true. I'm trying to dig up a diagram I thought I had showing where it comes from, but there is actually a link between the secant function and a secant line, and also the tangent function and a tangent line. –  ConMan Jun 29 at 23:47
    
In a right-angled triangle, $\cos\theta$ is the adjacent side divided by the hypotenuse. So, $\sec\theta$ is hypotenuse divided by adjacent. Don't know if that's the kind of explanation that would help. –  David Jun 29 at 23:49
    
@Rahul: You're wrong. It is completely the length of a segment of a secant line. That's exactly why it is so named. –  MPW Jun 29 at 23:51
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Wikipedia actually hosts a remarkably detailed diagram here: en.wikipedia.org/wiki/File:Circle-trig6.svg, which describes not only the (modern) trigonometric functions, but also other trigonometric functions that have fallen out of use, and some things that are not traditionally viewed as trigonometric functions (like the chord). is attached to their article about the versine. –  Euro Micelli Jun 30 at 1:33

3 Answers 3

up vote 13 down vote accepted

In this image, notice that as $\theta$ grows from $0$ to a right angle, the segment labeled $\sec\theta$ grows from $1$ to $+\infty$, and if $\theta$ were to go from $0$ down to minus a right angle, $\sec\theta$ would also go from $1$ up to $+\infty$.

On the other hand $\tan\theta$ would go from $0$ (not $1$) to $+\infty$ in one case, and from $0$ down to $-\infty$ in the other case.

Notice that the tangent is the length of a segment that is tangent (touching but not crossing) to the circle and the secant is the length of a segment that is secant to (cutting across) the circle. The words come from the Latin words tangere and secare, meaning to touch and to cut.

(The second image treats the three "co-" functions. I drew these with xfig several years ago and uploaded them to Wikipedia, where they are seen in this article.)

Some trigonometric functions

Here's an image treating the three "co-" functions

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A picture is worth a thousand words. That's exactly what I was trying to describe in my answer. Nice drawing, +1. –  MPW Jun 29 at 23:52
    
@MPW : Thank you. –  Michael Hardy Jun 29 at 23:55

The numeric values of the different trig functions are the lengths of the line segments in the diagram below, if the circle has radius $1$. This helped me a lot when trying to visualize what the numeric value of each trig function meant.

Relationship between trig functions

(Picture taken from Wikipedia.)

For Example, $\sec(\theta) = 2$ means that:

If one draws a tangent line from the point on the unit circle at $\theta$, the distance from the center of the unit circle to the intersection of the tangent line and the horizontal is $2$ units.

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Draw a ray from the origin making an angle of $\theta$ measured counterclockwise from the positive $x$-axis. You probably know that the point of intersection of this ray and the unit circle is the point $(\cos\theta,\sin\theta)$.

But if you also draw the vertical line tangent to the unit circle at $(1,0)$, then the intersection of the ray with this tangent line is the point $(1,\tan\theta)$. Moreover, the distance from the origin to this point is exactly $\sec\theta$.

In other words, $\sec\theta$ is the length of the secant line between the origin and the vertical tangent line, and $\tan\theta$ is the length of the vertical tangent line between the horizontal axis and the secant line.

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