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Unformatted text preview: ping an oriented arc around the Unit Circle to
ﬁnd coordinates on the Unit Circle, it should be clear that both the cosine and sine functions are
deﬁned for all real numbers t. In other words, the domain of f (t) = cos(t) and of g (t) = sin(t)
is (−∞, ∞). Since cos(t) and sin(t) represent x- and y -coordinates, respectively, of points on the
Unit Circle, they both take on all of the values between −1 an 1, inclusive. In other words, the
range of f (t) = cos(t) and of g (t) = sin(t) is the interval [−1, 1]. To summarize:
Theorem 10.5. Domain and Range of the Cosine and Sine Functions:
• The function f (t) = cos(t) • The function g (t) = sin(t) – has domain (−∞, ∞) – has domain (−∞, ∞) – has range [−1, 1] – has range [−1, 1] 630 Foundations of Trigonometry 1
Suppose, as in the Exercises, we are asked to solve an equation such as sin(t) = − 2 . As we have
already mentioned, the distinction between t as a real number and as an angle θ = t radians is often
blurred. Indeed, we solve sin(t) = − 1 in the exact same manner13 as we did in Example 10.2.5
number 2. Our solution is only cosmetically diﬀerent in that the variable used is t ra...
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