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**Unformatted text preview: **ned by tan(θ) = While we left the history of the name ‘sine’ as an interesting research project in Section 10.2, the
names ‘tangent’ and ‘secant’ can be explained using the diagram below. Consider the acute angle θ
below in standard position. Let P (x, y ) denote, as usual, the point on the terminal side of θ which
lies on the Unit Circle and let Q(1, y ) denote the point on the terminal side of θ which lies on the
vertical line x = 1.
y Q(1, y ) = (1, tan(θ)) 1
P (x, y ) θ
O A(x, 0) B (1, 0) x 636 Foundations of Trigonometry The word ‘tangent’ comes from the Latin meaning ‘to touch,’ and for this reason, the line x = 1
is called a tangent line to the Unit Circle since it intersects, or ‘touches’, the circle at only one
point, namely (1, 0). Dropping perpendiculars from P and Q creates a pair of similar triangles
y
1
∆OP A and ∆OQB . Thus y = x which gives y = x = tan(θ), where this last equality comes from
y
applying Deﬁnition 10.2. We have just shown that for acute angles θ, tan(θ) is the y -coordinate...

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