Unformatted text preview: adrant IV angle, we also know x > 0 and y < 0. Viewing −4 = −1 , we may choose5 x = 4
√
and y = −1 so that r = x2 +√ 2 = (4)2 + (−1)2 = √ 17. Applying Theorem 10.9 once
y
√
√
17
17
17
more, we ﬁnd cos(θ) = √4 = 4 17 , sin(θ) = − √1 = − 17 , sec(θ) = 4 , csc(θ) = − 17,
17
17
and tan(θ) = − 1 .
4
We may also specialize Theorem 10.9 to the case of acute angles θ which reside in a right triangle,
as visualized below. c
b
θ
a
Theorem 10.10. Suppose θ is an acute angle residing in a right triangle. If the length of the side
adjacent to θ is a, the length of the side opposite θ is b, and the length of the hypotenuse is c,
then
b
c
c
a
tan(θ) =
sec(θ) =
csc(θ) =
cot(θ) =
a
a
b
b
The following example uses Theorem 10.10 as well as the concept of an ‘angle of inclination.’ The
angle of inclination (or angle of elevation) of an object refers to the angle whose initial side is some
kind of baseline (say, the ground), and whose terminal side is the li...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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