TRIGONOMETRY OUTCOMES

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2 TRIGONOMETRY OUTCOMES C10. Solve problems involving limits of trigonometric functions. C11. Apply derivatives of trigonometric functions. C12. Solve problems involving inverse trigonometric functions.

3 LIMITS OF TRIGONOMETRIC FUNCTIONS We have already worked with limits of algebraic functions. This unit begins with an exploration of the limits of trigonometric functions and expressions. Pre-requisite Knowledge In Mathematics 3200 you graphed and analyzed the trigonometric functions sine, cosine and tangent.

4 LIMITS OF TRIGONOMETRIC FUNCTIONS Graph and verify using direct substitution the behavior of the following: lim sinx x 0 y x

5 LIMITS OF TRIGONOMETRIC FUNCTIONS Graph and verify using direct substitution the behavior of the following: lim cosx x 0 y x

6 EXAMPLES: (USING A TABLE OF VALUES) sin Determine: sin lim lim (radians) sin (radians) sin sin lim 1 0 sin lim 1 0 6

7 IS THE RESULT THE SAME IF WE USE DEGREES? sin Determine: lim 0 (degrees) sin (degrees) sin

8 Using a graphical approach, examine the behavior of the following: lim x 0 sin x x Radians y Degrees y x

9 Using a graphical approach, examine the behavior of the following: lim x 0 cosx 1 x Also Note: lim x 0 1 cosx x y x

10 TRIGONOMETRIC LIMITS Strategies to evaluate trigonometric limits: Factoring Rationalizing Rewriting the trigonometric expression using trigonometric identities

11 EXAMPLES Use previous limit 1. sin lim 1 x 0 sin2 lim 0 sin2 2 sin2 lim 2lim Also, as 0 so does 2 0 x to help determine the following: x This must match the previous limit. This means every part of the limit must have the same term: 2. sin2 2 lim (1) 2 11

12 EXAMPLES Use previous limits 2. sin3x x lim x 0 4 Also, as sinx lim 1 x x 0 1 sin3x 3 lim 4 x 0 x 3 x 0 so does 3x 0 to help determine the following: 3 sin3x lim 4 x 0 3x 3 sin3x lim 4 3x 0 3x 3 3 (1)

13 lim x 0 2 x sin2x x x 0 sin2x lim x (2 x 5) sin2x 1 lim x 0 x 2x 5 sin2x 1 lim lim x0 x x0 2 x

14 4. lim x 0 cosx tanx x 14

15 5. lim x 0 tanx x 15

16 6. lim x 0 (1 cos x ) x 2 16

17 7. 1 tan x lim sin x cos x x 4 17

18 8. sin5x x lim x 0 sin2 Hint: You need to get sinx/x in the top and bottom lim x 0 2 5x sin5 x 5x x sin2 x 2x sin5x lim 5 x 5x 2x sin2 x 2x x 0 5 sin5x 5x x 2 lim x 0 sin2 2x lim sin5x x sin2 x x 5 5x lim 2x

19 19

20 20

22 DERIVATIVES FROM GRAPHS OR LINK Tec Animations 2 Derivatives Visual 2.3

24 DERIVATIVES OF TRIG USING DEFINITION OF DERIVATIVE Recall two limits used in trig limits: lim h0 sinh h lim h0 cosh 1 h

25 FIND THE DERIVATIVE OF F(X) = SIN X USING THE DEFINITION OF DERIVATIVE f ( x h) f ( x ) f( x) lim h0 h d sin( x h) sinx sinx lim dx h0 h We need the Sine Addition Rule ( sin x h sinx cosh sinh cosx sinx cosh sinh cosx sinx lim h0 h sinx coshsin x sinh cosx lim h0 h sin x (cosh1) sinh cosx lim h0 h

26 sin x (cosh 1) sinh cosx lim lim h0 h h0 h (cosh 1) sinh sinx lim cosx lim h0 h h0 h Therefore: ( x( sin x 0 cos 1 cosx d sinx cosx dx

27 ALSO: d cosx sinx dx You get to prove this on your next assignment!! NOTE: cos x h cos x cosh sin x sinh (

28 FIND: d tanx dx Hint: Use the quotient rule!

29 NOTE: d cotx dx Find d secx dx cscx

30 d NOTE: cscx cscx cotx dx

31 RECAP: d sinx dx d tanx dx d secx dx d cosx dx d cotx dx d cscx dx

32 EXAMPLE: DIFFERENTIATE 1.A) f(x) = 3 sinx + 4 tanx B) f(x) = x 3 sec x

33 C) y cosx x

35 1 sinx D) y cosx

36 EXAMPLES: FIND d dx E) 2 2 cosx cos x

37 EXAMPLES: FIND d dx F) 3 sin x

38 EXAMPLES: FIND d dx G) sec( sinx

39 EXAMPLES: FIND d dx H) 2 tanx

40 EXAMPLES: FIND I) d cot x 2 dx x 1

41 EXAMPLES: FIND J) d cos( x 2) dx 2 3

42 EXAMPLES: FIND d dx K) ( 2 3 cos( x 2)

43 d 2 2. A) SHOW cos x dx sin2x

44 2. B) Show that for 3 f ( x ) cscx 2csc x f ( x ) cotx cscx

45 3. Find the equation of the tangent line to the graph of at x = 0 y sinx

46 4. Find the equations of the tangent and the normal lines to the graph of y cosx at x 6

47 5. Find y for cosx cos y 1

48 PAGE DO a, b, e, f, h, j, k, l, m, o, p, r, t

50 TANGENT LINES Determine the equation of the tangent line of the relation sin(xy - y 2 ) = x 2-1 at the point (1,1).

51 RELATED RATES 1. Two sides of a triangle measure 5 m and 8 m in length. The rad angle between them is increasing at a rate of. How 45 sec fast is the length of the third side changing when the contained angle is? 3 First step is to sketch a diagram and label the information. What function relates the angle and the opposite side? The Law of Cosines!

53 2. The sun is setting. A 10 m tree is casting a shadow whose length at 7:00 pm was 10 3 m. If the shadow is increasing at a rate of 3 cm/min how fast is the sun setting at 7:00 pm.

54 3. A kite 40 m above the ground moves horizontally at the rate of 3 m/s. At what rate is the angle between the string and the horizontal decreasing when 80 m of string has been let out?

55 TEXT Page # 47, 51, 64, 65, , # 22, 28, 29

56 DO NUMBER 68, Page 122

57 OPTIMIZATION PROBLEMS 1. The position of a particle as it moves horizontally is described by the equation s = 2sint - cost, 0 t 2π, where s is displacement in metres and t is the time in seconds. Find the absolute maximum and minimum displacements. Recall velocity is the rate of change in displacement with respect to time. Also the absolute maximum or minimum (displacements) occur when the derivative (velocity) equals zero or at the endpoints.

59 2. Triangle ABC is inscribed in a semicircle with diameter BC = 10 cm. Find the value of angle B that produces the triangle of maximum area. Any triangle inscribed in a semi-circle, will have a right angle. If either angle B or C are 90 o then the triangle would not be inscribed:

60 That means that angle A is 90 o so BC must be the hypotenuse. We can use trig ratios to find length of the height and base of the triangle. b sinb b 10 sinb 10 c cosb c 10cosB 10 1 Area base height 2 1 A 10sin B 10cos B 2 A 50sin B cosb ( (

61 A 50sin B cosb Find the Absolute Max for A A 50(cos B cosb sinb sin B ) A B B (cos sin ) ( A 50(cos2 B) 50(cos2 B) 0 cos2b 0 1 2B cos (0) 2B B 2 4 A 0 A B B (cos sin ) (cos B sin B) cos B sin B cos B sin 2 sin B 1 2 cos B B

62 B B 2 tan B 1 tanb tan (1) 4 B 1 tan ( 1) B 4 Since negative angles are non-admissible answers for the context of this problem the answer is B 4

64 Text Page 151 # 52. Page 183 # 56

66 Explain the relationship between the primary trigonometric functions and the inverse trigonometric functions. Explain why trigonometric functions have their domains restricted to create inverse trigonometric. Sketch the graph of an inverse trigonometric function Determine the exact value of an expression involving an inverse trigonometric function. Derive the inverse trigonometric derivatives. Determine the derivative of an inverse trigonometric function. Solve problems involving the derivative of an inverse trigonometric function.

67 INVERSES RECALL?? A relation that is symmetric to a function across the line y = x is said to be the inverse of the function. If the relation is a function it is called an inverse function and is given the notation f -1 (x). A test for symmetry in the line y = x is to interchange x and y. If the relation is the same then the relation is symmetric with respect to the y = x.

68 FINDING INVERSES USING f ( x ) Let f(x) = y x Interchange x and y 2 AS AN EXAMPLE. y x Solve for y. y x This new y is an inverse relation to f(x), but it may not be an inverse function. 2 x y 2

69 f ( x ) x 2 y y x x Is y x a function or relation? Relation. It does not pass the vertical line test y x It may be necessary to restrict the domain of the original function, f(x), if f(x) does not pass the horizontal line test.

70 HORIZONTAL LINE TEST A function passes the horizontal line test if any horizontal line crosses (or intersects) the graph of a function in, at most, one place. A function that passes the horizontal line test is said to be one-to-one. That is for every y-value in the range there corresponds exactly one x-value in the domain.

71 f ( x ) x 2 y x Does the horizontal line test? No. f ( x ) x Thus the inverse is not a function. 2 pass So how can we restrict the domain of the original function, f(x), such that the inverse of f(x) is a function?

72 THERE ARE 3 CONDITIONS THAT MUST BE MET WHEN RESTRICTING THE DOMAIN TO OBTAIN AN INVERSE FUNCTION 1. The restricted function must pass the horizontal line test. Resulting in an inverse that is a function 2. We want to included the complete set of y-values. (ALL of the RANGE) 3. The restricted functions contain the centre part of graph of the original function. For parabolas the vertex must be included. If the vertex is (h, k) then we can say x h x h OR For periodic functions (Trig Functions) the y-intercept must be included.

73 RESTRICTION FOR x 0 f ( x ) x Lets take since the vertex is (0, 0). 2 This gives for the inverse. y y 0 x y f 1 ( x ) x x

74 NOTE: The notation used for inverse functions is In this case this means the inverse function of f(x). It does NOT mean the reciprocal of f(x). f ( x) 1 1 f( x) f 1 ( x)

75 EXAMPLE: FIND THE INVERSE FUNCTION OF f(x) = x 2-4

76 What restriction do we take? 0, 4 ( What is the vertex? x 0 x 0 f 1 ( x ) x 4 f 1 ( x ) x 4

77 VERIFY: 1 ( ( )) f f x x 1 f ( f ( x )) x NOTE: This is a test for inverses.

78 FIND THE INVERSE FUNCTION OF F(X) = X 2 + 6X

79 What restriction do we take? What is the vertex? Recall h= b/(2a) x 3 x 3

80 WARM UP: WHICH OF THE FOLLOWING WILL HAVE AN INVERSE FUNCTION WITHOUT MAKING A RESTRICTION?

81 INVERSE TRIG Consider y = sin x Does this graph have an inverse function? NO! Why? It fails the horizontal line test. We must restrict the domain so that the inverse is a function.

82 Recall: There are 3 things to keep in mind when restricting the domain: 1) The restricted function must pass the horizontal line test. 2) We want to included the complete set of y-values. (ALL of the RANGE) 3) We want to include the centre part of the graph. It was the vertex for quadratic functions This is the y-intercept for Trig functions

83 WHAT IS THE RESTRICTION APPLIED THE SINE FUNCTION? Restricted Domain Range: x y 1 This restricted function is called the principal function can is represented by: y = Sin x The capital S means the function is restricted.

84 FIND THE EQUATION OF THE INVERSE SINE FUNCTION. o y = Sin x o Interchange x and y o x = Sin y o We can only solve for y by introducing a new function. o y = sin -1 x o Note: sin ( x ) 1 1 sin( x ) o"-1" is not an exponent, it is a notation that is used to denote inverse trigonometric functions. o If the "-1 represented an exponent it would be written as ( sin( x ) 1 1 sin( x )

85 o There are other notations used for inverse trigonometric functions that avoids this ambiguity o The inverse sine function can be written as y = sin -1 x or y = arcsin x or y = invsin x or y = asn x o Regardless of what it is called they each mean that y is the ANGLE whose sin is x (ie Sin y = x)

86 Domain of y = arcsin x: 1 x 1 Range is: y 2 2

87 GRAPH OF Y = ARCSIN X Where are the Angles on the unit circle that result from y = arcsin x?... Quadrants I and IV

88 SOLVE: IN EXACT RADIANS IF POSSIBLE 1. arcsin (0.3) 2. sin x = sin 1 2 2

89 FIND THE INVERSE OF COSINE Restricted Domain 0 x Range: 1 y 1 y = Cos x

90 FIND THE EQUATION OF THE INVERSE COSINE FUNCTION. y = Cos x Interchange x and y x = Cos y We can only solve for y by introducing a new function. y = cos -1 x or y = arccos x y is the ANGLE whose cosine is x (ie Cos y = x)

91 Domain of y = arccos x: 1 x 1 Range is: 0 y

92 GRAPH OF Y = ARCCOS X Where are the angles that result from y = arccos x?... Quadrants I and II

93 SOLVE: IN EXACT RADIANS IF POSSIBLE 1. arccos (0.5) 2. cos cosx arccos (3)

94 FIND THE EXACT VALUES FOR: A) arcsin 1 B) cos(arcsin 1) C) tancos 1 2 2

95 D) sin arccos 4 5

96 E) sin(arccos z )

97 FIND THE INVERSE OF TANGENT Restricted Domain x 2 2 Range: y R y = Tan x What is the equations of the asymptotes? x and x 2 2

98 FIND THE EQUATION OF THE INVERSE TANGENT FUNCTION. y = Tan x Interchange x and y x = Tan y We can only solve for y by introducing a new function. y = tan -1 x or y = arctan x y is the ANGLE whose tangent is x (ie Tan y = x)

99 Domain of y = arctan x: x R Range is: y 2 2 What is the equations of the asymptotes? y and y 2 2

100 GRAPH OF Y = ARCTAN X Where are the angles that result from y = arctan x?... Quadrants I and IV

101 SOLVE TO THE NEAREST DEGREE 1. arctan (-1) 2. tan arctan arctan (-0.1) 5. arctan(tan45 o ) 6. arctan(tan 135 o )

102 FIND THE EXACT VALUES FOR: A) tan(arcsin 1) B) cos(arctan(1)) C) sintan

103 D) 3 cosarctan 8

104 E) cos(arctan m)

105 Sketch the graph of y = arcsec x. State the domain:... Range Equation of asymptotes

106

107 INVERSE TRIG DERIVATIVES To find the derivative of y = sin -1 x we implicitly differentiate sin y = x siny x Recall y = sin -1 x = arcsin x y is the angle whose sine is x, or sin y = x. 1 x y 2 1 x

108 DIFFERENTIATE Y = COS -1 X

109 DIFFERENTIATE Y = TAN -1 X

110 DIFFERENTIATE Y = COT -1 X

111 DIFFERENTIATE Y = SEC -1 X

112 DIFFERENTIATE Y = CSC -1 X

113

114 dy EXAMPLES. FIND dx 1. y arcsin ( 2 x

115 2 2. y arccos ( x

116 3. y arctan ( cosx

117 1 4. y sec ( arctan x

118 sin ( 1 x

119 x cos ( ) x

120

121

122 Lets do #3. Determine the equation of a tangent line to a curve represented by an inverse trigonometric function. 1 x 2 f ( x x sin 16 x at x = 2 4

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained.

Using this definition, it is possible to define an angle of any (positive or negative) measurement by recognizing how its terminal side is obtained. Angle in Standard Position With the Cartesian plane, we define an angle in Standard Position if it has its vertex on the origin and one of its sides ( called the initial side ) is always on the positive