Pre-Calc Exam Notes 36

Pre-Calc Exam Notes 36 - 36 Chapter 1 • Right Triangle...

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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. y (− x , y ) ( x , y) ◦ 180 − θ r r θ x 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 function value. Example 1.27 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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