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Unformatted text preview: 36 Chapter 1 • Right Triangle Trigonometry §1.5 Notice that reﬂection around the y-axis is equivalent to reﬂection around the x-axis (θ →
−θ ) followed by a rotation of 180◦ (−θ → −θ + 180◦ = 180◦ − θ ), as in Figure 1.5.7.
(− x , y ) ( x , y) ◦ 180 − θ r r
r −θ ( x , − y)
Figure 1.5.7 Reﬂection of θ around the y-axis = 180◦ − θ It may seem that these geometrical operations and formulas are not necessary for evaluating the trigonometric functions, since we could just use a calculator. However, there are
two reasons for why they are useful. First, the formulas work for any angles, so they are
often used to prove general formulas in mathematics and other ﬁelds, as we will see later
in the text. Second, they can help in determining which angles have a given trigonometric
Find all angles 0◦ ≤ θ < 360◦ such that sin θ = −0.682. ✄ sin
Solution: Using the ✂ −1 ✁button on a calculator with −0.682 as the input, we get θ = −43◦ , which
is not between 0◦ and 360◦ .7 Since θ = −43◦ is in QIV, its reﬂection 180◦ − θ around the y-axis will be
in QIII and have the same sine value. But 180◦ − θ = 180◦ − (−43◦ ) = 223◦ (see Figure 1.5.8). Also, we
know that −43◦ and −43◦ + 360◦ = 317◦ have the same trigonometric function values. So since angles
in QI and QII have positive sine values, we see that the only angles between 0◦ and 360◦ with a sine
of −0.682 are θ = 223◦ and 317◦ . y x
180◦ − θ = 223◦ r
(− x , y ) r θ = −43◦ ( x , y) Figure 1.5.8 Reﬂection around the y-axis: −43◦ and 223◦ ✄ 7 In Chapter 5 we will discuss why the sin−1 button returns that value.
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