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Trigonometric Functions In Exercises 1 - 4, convert from radians to degrees or degrees to radians. p 1. 2. - 2.5 3 3. - 40° 4. 45° In Exercises 5 - 7, solve the equation graphically in the given interval. 5. sin x = 0.6, 0 £ x < 2p 6. cos x = - 0.4, 0 £ x < 2p p 3p 7. tan x = 1, - £ x< 2 2 Sli de 12 8. Show that f (x )= 2 x 2 - 3 is an even function. Explain why its graph is symmetric about the y -axis. 9. Show that f (x )= x 3 - 3 x is an odd function. Explain why its graph is symmetric about the origin. 10. Give one way to restrict the domain of the function f (x )= x 4 - 2 to make the resulting function one-to-one. Sli de 13 In Exercises 1 - 4, convert from radians to degrees or degrees to radians. 1. p 3 60° 2. - 2.5 - 143.24° 2p p 4. 45° 9 4 In Exercises 5 - 7, solve the equation graphically in the 3. - 40° - given interval. 5. sin x = 0.6, 6. cos x = - 0.4, 7. tan x = 1, 0 £ x < 2p 0 £ x < 2p p 3p - £ x< 2 2 x » 0.6435, 2.4981 x » 1.9823, 4.3009 x » 0.7854, 3.9270 Sli de 14 8. Show that f (x )= 2 x 2 - 3 is an even function. explain why its graph is symmetric about the y -axis. 2 f (- x )= 2(- x ) - 3= 2 x 2 - 3= f (x ) The graph is symmetric about the y -axis because if a point (a, b) is on the graph, then so is the point (- a, b). Sli de 15 9. Show that f (x )= x 3 - 3 x is an odd function. Explain why its graph is symmetric about the origin. 3 f (- x )= (- x ) - 3(- x )= - x 3 + 3 x = - f (x ) The graph is symmetric about the origin because if a point (a, b)is on the graph, then so is the point (- a, - b). 10. Give one way to restrict the domain of the function f (x)= x 4 - 2 to make the resulting function one-to-one. x³ 0 Sli de 16 Radian Measure Graphs of Trigonometric Functions Periodicity Even and Odd Trigonometric Functions Transformations of Trigonometric Graphs Inverse Trigonometric Functions …and why Trigonometric functions can be used to model periodic behavior and applications such as musical notes. Sli de 17 The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Sli de 18 An angle of measure θ is placed in standard position at the center of circle of radius r, Sli de 19 The six basic trigonometric functions of q are defined as follows: y sine: sin q = r x cosine: cos q = r y tangent: tan q = x r y r secant: sec q = x x cotangent: cot q = y cosecant: csc q = Sli de 110 When we graph trigonometric functions in the coordinate plane, we usually denote the independent variable (radians) by x instead of θ . Angle Convention: Use Radians From now on in this book, it is assumed that all angles are measured in radians unless degrees or some other unit is stated explicitly. When we talk about the angle p p p we mean radians ( which is 60°), not degrees. 3 3 3 When you do calculus, keep your calculator in radian mode. A function f (x ) is periodic if there is a positive number p such that f (x + p )= f (x ) for every value of x. The smallest value of p is the period of f . The functions cos x, sin x, sec x and csc x are periodic with period 2p . The functions tan x and cot x are periodic with period p . The graphs of cos x and sec x are even functions because their graphs are symmetric about the yaxis. The graphs of sin x, csc x, tan x and cot x are odd functions. y = cos x y = sin x Sli de 114 Show that csc x is an odd function. 1 1 csc (- x)= = - csc x sin (- x ) - sin x Sli de 115 The rules for shifting, stretching, shrinking and reflecting the graph of a function apply to the trigonometric functions. Vertical stretch or shrink Reflection about x-axis Vertical shift y = a f (b(x + c))+ d Horizontal stretch or shrink Reflection about the y-axis Horizontal shift y = a f (b(x + c))+ d y = a f (b(x + c))+ d Determine the period, domain, range and draw the graph of y = - 2sin (4 x + p ) None of the six basic trigonometric functions graphed above is one-to-one. These functions do not have inverses. However, in each case, the domain can be restricted to produce a new function that does have an inverse. The domains and ranges of the inverse trigonometric functions become part of their definitions. Slide 1- 19 The graphs of the six inverse trigonometric functions are shown here. Slide 1- 21 Example 7 pg.50, # 27 pg. 53 Example 8 pg.51 # 31 Homework for the week: Worksheets, # 9-12 pg. 52