Pre-Calc Exam Notes 33

# Pre-Calc Exam Notes 33 - Rotations and Reections of Angles...

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Rotations and Reflections of Angles Section 1.5 33 Example1.26 Recall that any nonvertical line in the xy -coordinate plane can be written as y = mx + b , where m is the slope of the line (defined as m = rise run ) and b is the y -intercept , i.e. where the line crosses the y - axis (see Figure 1.5.2(a)). We will show that the slopes of perpendicular lines are negative reciprocals. That is, if y = m 1 x + b 1 and y = m 2 x + b 2 are nonvertical and nonhorizontal perpendicular lines, then m 2 =− 1 m 1 (see Figure 1.5.2(b)). x y 0 y = mx + b b run rise m = rise run (a) Slope of a line x y 0 y = m 1 x + b 1 b 1 y = m 2 x + b 2 b 2 (b) Perpendicular lines Figure 1.5.2 First, suppose that a line y = mx + b has nonzero slope. The line crosses the x -axis somewhere, so let θ be the angle that the positive x -axis makes with the part of the line above the x -axis, as in Figure 1.5.3. For m > 0 we see that θ is acute and tan θ = rise run = m . x y 0 θ y = mx + b b run > 0 rise > 0 (a) m > 0, θ acute, rise > 0 x y 0 θ y = mx + b b run > 0 rise < 0 (b) m < 0, θ obtuse, rise < 0 Figure 1.5.3 If m < 0, then we see that θ is obtuse and the rise is negative. Since the run is always positive,
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