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the point on the terminal side of θ which lies on the line x = 1 which is tangent to the Unit Circle.
Now the word ‘secant’ means ‘to cut’, so a secant line is any line that ‘cuts through’ a circle at two
points.1 The line containing the terminal side of θ is a secant line since it intersects the Unit Circle
in Quadrants I and III. With the point P lying on the Unit Circle, the length of the hypotenuse
of ∆OP A is 1. If we let h denote the length of the hypotenuse of ∆OQB , we have from similar
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triangles that h = x , or h = x = sec(θ). Hence for an acute angle θ, sec(θ) is the length of the line
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segment which lies on the secant line determined by the terminal side of θ and ‘cuts oﬀ’ the tangent
line x = 1. Not only do these observations help explain the names of these functions, they serve as
the basis for a fundamental inequality needed for Calculus which we’ll explore in the Exercises.
Of the six circular functions, only cosine and sine are deﬁned for all angles. Since cos(θ) =...

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