Preliminaries 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2 LINES AND FUNCTIONS 0.3 GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS 0.4 TRIGONOMETRIC FUNCTIONS 0.5 TRANSFORMATIONS OF FUNCTIONS © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 2

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.1 A function f is periodic of period T if f (x + T ) = f (x) for all x such that x and x + T are in the domain of f. The smallest such number T > 0 is called the fundamental period © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 3

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS The Unit Circle, Sine, and Cosine Measure θ in radians, given by the length of the arc indicated in the figure. Define sin θ to be the y-coordinate of the point on the circle and cos θ to be the x-coordinate of the point. It follows that sin θ and cos θ are defined for all values of θ, so that each has domain −∞ < θ < ∞, while the range for each of these functions is the interval [−1, 1]. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 4

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.1 Unless otherwise noted, we always measure angles in radians. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 5

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.1 The functions f (θ) = sin θ and g(θ) = cos θ are periodic, of period 2π. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 6

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.1 Solving Equations Involving Sines and Cosines Find all solutions of the equations (a) 2 sin x − 1 = 0 and (b) cos2 x − 3 cos x + 2 = 0. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 7

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.1 Solving Equations Involving Sines and Cosines (a) 2 sin x − 1 = 0 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 8

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.1 Solving Equations Involving Sines and Cosines (b) cos2 x − 3 cos x + 2 = 0 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 9

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 10

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 11

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 12

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 13

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.2 Altering Amplitude and Period Graph y = 2 sin x and y = sin 2x, and describe how each differs from the graph of y = sin x. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 14

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.2 Altering Amplitude and Period © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 15

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS Amplitude and Frequency A is the amplitude f is the frequency: p is the period: p = 1/f © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 16

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 17

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.6 Writing Combinations of Sines and Cosines as a Single Sine Term Prove that 4 sin x + 3 cos x = 5 sin(x + β) for some constant β and estimate the value of β. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 18

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.6 Writing Combinations of Sines and Cosines as a Single Sine Term This will occur if we can choose a value of β so that cos β = 45 and sin β = 35. By the Pythagorean Theorem, this is possible only if sin2 β + cos2 β = 1. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 19

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.6 Writing Combinations of Sines and Cosines as a Single Sine Term For the moment, the only way to estimate β is by trial and error. Using your calculator or computer, you should find that one solution is β ≈ 0.6435 radians (about 36.9 degrees). © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 20

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.8 Finding the Height of a Tower A person 100 feet from the base of a radio tower measures an angle of 60o from the ground to the top of the tower. Find the height of the tower. © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 21

TRIGONOMETRIC FUNCTIONS 0.4 TRIGONOMETRIC FUNCTIONS 4.8 Finding the Height of a Tower Using the similar triangles: © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Slide 22