In geometry, a secant is a line that intersects a circle at two points (from the Latin word secare, "to cut") Compare to tangent.
In trigonometry, the secant (sec) function is so named because of this geometric meaning. If you draw a unit circle on a pair of coordinate axes and then add a line segment out from its center, that line will strike the unit circle at one point. The xcoordinate of that point is the cosine of the angle formed by the line segment.
cosine
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If you extend this line segment beyond the unit circle, far enough that you can drop a vertical line that is tangent to the circle, you create a new triangle which is similar to the one inside the unit circle:
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secant < /
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Because the triangles are similar, the ratios of their sides are equal. Since we are dealing with a unit circle, its radius is 1. The hypotenuse of the smaller triangle and the base of the larger triangle are both radii. Therefore, the base of the smaller triangle (the cosine of the angle) divided by its hypotenuse (1) equals the base of the larger triangle (1) divided by its hypotenuse (call it 'h'):
cosine 1
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1 h
So the length of the large triangle's hypotenuse, which is part of a secant line through the circle, equals the reciprocal of the cosine.