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Unformatted text preview: Rotations and Reﬂections of Angles • Section 1.5 35 Applying this to angles, we see that the reﬂection of an angle θ around the x-axis is the
angle −θ , as in Figure 1.5.6.
( x , y) ( x , y) r r θ θ x x −θ
( x , − y) ( x , − y) (a) QI and QIV −θ (b) QII and QIII Figure 1.5.6 Reﬂection of θ around the x-axis So we see that reﬂecting a point ( x, y) around the x-axis just replaces y by − y. Hence:
sin (−θ ) = − sin θ (1.13) cos (−θ ) = cos θ (1.14) tan (−θ ) = − tan θ (1.15) Notice that the cosine function does not change in formula (1.14) because it depends on x,
and not on y, for a point ( x, y) on the terminal side of θ .
In general, a function f ( x) is an even function if f (− x) = f ( x) for all x, and it is called an
odd function if f (− x) = − f ( x) for all x. Thus, the cosine function is even, while the sine and
tangent functions are odd.
Replacing θ by −θ in formulas (1.4)−(1.6), then using formulas (1.13)−(1.15), gives:
sin (90◦ − θ ) = cos θ (1.16) cos (90◦ − θ ) = sin θ (1.17) tan (90◦ − θ ) = cot θ (1.18) Note that formulas (1.16)−(1.18) extend the Cofunction Theorem from Section 1.2 to all θ ,
not just acute angles. Similarly, formulas (1.10)−(1.12) and (1.13)−(1.15) give:
sin (180◦ − θ ) = sin θ (1.19) cos (180◦ − θ ) = − cos θ (1.20) tan (180◦ − θ ) = − tan θ (1.21) ...
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