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Unformatted text preview: Trigonometric Functions of Any Angle • Section 1.4 25 We can now deﬁne the trigonometric functions of any angle in terms of Cartesian coordinates. Recall that the x y-coordinate plane consists of points denoted by pairs ( x, y) of
real numbers. The ﬁrst number, x, is the point’s x coordinate, and the second number, y, is
its y coordinate. The x and y coordinates are measured by their positions along the x-axis
and y-axis, respectively, which determine the point’s position in the plane. This divides the
x y-coordinate plane into four quadrants (denoted by QI, QII, QIII, QIV), based on the signs
of x and y (see Figure 1.4.3(a)-(b)).
y>0 0 (2, 3)
(−3, 2) x x 0 QIV
y<0 (−2, −2) (a) Quadrants I-IV (3, −3) (b) Points in the plane y ( x , y) r
θ x 0 (c) Angle θ in the plane Figure 1.4.3 x y-coordinate plane Now let θ be any angle. We say that θ is in standard position if its initial side is the
positive x-axis and its vertex is the origin (0, 0). Pick any point ( x, y) on the terminal side of
θ a distance r > 0 from the origin (see Figure 1.4.3(c)). (Note that r = x2 + y2 . Why?) We
then deﬁne the trigonometric functions of θ as follows:
sin θ = y
r cos θ = x
r tan θ = y
x (1.2) csc θ = r
y sec θ = r
x cot θ = x
y (1.3) As in the acute case, by the use of similar triangles these deﬁnitions are well-deﬁned (i.e.
they do not depend on which point ( x, y) we choose on the terminal side of θ ). Also, notice
that | sin θ | ≤ 1 and | cos θ | ≤ 1, since | y | ≤ r and | x | ≤ r in the above deﬁnitions. ...
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