June 9 Math 1113 sec 002 Summer 2014

Transcription

1 June 9 Math 1113 sec 002 Summer 2014 Section 6.5: Inverse Trigonometric Functions Definition: (Inverse Sine) For x in the interval [ 1, 1] the inverse sine of x is denoted by either and is defined by the relationship sin 1 (x) or arcsin(x) y = sin 1 (x) x = sin(y) where π 2 y π 2. The Domain of the Inverse Sine is 1 x 1. The Range of the Inverse Sine is π 2 y π 2. () June 6, / 68

2 Notation Warning! Caution: We must remember not to confuse the superscript 1 notation with reciprocal. That is sin 1 (x) 1 sin(x). If we want to indicate a reciprocal, we should use parentheses or trigonometric identities 1 sin(x) = (sin(x)) 1 or write 1 sin(x) = csc(x). () June 6, / 68

3 Some Inverse Sine Values We can build a table of some inverse sine values by using our knowledge of the sine function. x sin(x) π 2 1 π π π π π 2 1 x sin 1 (x) 1 π π π π 4 π 3 1 π 2 () June 6, / 68

4 Conceptual Definition We can think of the inverse sine function in the following way: sin 1 (x) is the angle between π 2 and π 2 whose sine is x. () June 6, / 68

5 Example Evaluate each expression exactly. ( ) 1 (a) sin 1 2 ) 3 (b) sin ( 1 2 (c) sin 1 (1) () June 6, / 68

6 The Graph of the Arcsine Figure: Note that the domain is 1 x 1 and the range is π 2 y π 2. () June 6, / 68

7 Function/Inverse Function Relationship For every x in the interval [ 1, 1] ( ) sin sin 1 (x) = x For every x in the interval [ π 2, π 2 ] sin 1 (sin(x)) = x Remark 1: If x > 1 or x < 1, the expression sin 1 (x) is not defined. Remark 2: If x > π 2 or x < π 2, the expression sin 1 (sin(x)) IS defined, but IS NOT equal to x. () June 6, / 68

8 Example Evaluate each expression if possible. If it is not defined, give a reason. [ ( )] 1 (a) sin sin 1 2 (b) sin 1 (3) () June 6, / 68

9 (c) [ ( π )] sin 1 sin 8 (d) sin 1 [sin ( )] 4π 3 () June 6, / 68

10 The Inverse Cosine Function Figure: To define an inverse cosine function, we start by restricting the domain of cos(x) to the interval [0, π] () June 6, / 68

11 The Inverse Cosine Function (a.k.a. arccosine function) Definition: For x in the interval [ 1, 1] the inverse cosine of x is denoted by either and is defined by the relationship cos 1 (x) or arccos(x) y = cos 1 (x) x = cos(y) where 0 y π. The Domain of the Inverse Cosine is 1 x 1. The Range of the Inverse Cosine is 0 y π. () June 6, / 68

12 The Graph of the Arccosine Figure: Note that the domain is 1 x 1 and the range is 0 y π. () June 6, / 68

13 Function/Inverse Function Relationship For every x in the interval [ 1, 1] ( ) cos cos 1 (x) = x For every x in the interval [0, π] cos 1 (cos(x)) = x Remark 1: If x > 1 or x < 1, the expression cos 1 (x) is not defined. Remark 2: If x > π or x < 0, the expression cos 1 (cos(x)) IS defined, but IS NOT equal to x. () June 6, / 68

14 Some Inverse Cosine Values We can build a table of some inverse sine values by using our knowledge of the sine function. x cos(x) 0 1 π 6 π π 2 0 2π π π 1 x cos 1 (x) π 6 π 4 π 0 2 2π 1 3 5π 6 π () June 6, / 68

15 Conceptual Definition We can think of the inverse cosine function in the following way: cos 1 (x) is the angle between 0 and π whose cosine is x. () June 6, / 68

16 Examples Evaluate each expression exactly. ( (a) cos 1 1 ) 2 (b) cos 1 (0) ( (c) cos 1 1 ) 2 () June 6, / 68

17 Examples Evaluate each expression if possible. If undefined, state a reason. [ ( )] 1 (a) cos cos 1 4 (b) [ ] cos cos 1 (6) () June 6, / 68

18 ( )] 7π (c) cos [cos 1 8 (d) [ ( cos 1 cos π )] 4 () June 6, / 68

19 The Inverse Tangent Function Figure: To define an inverse tangent function, we start by restricting the domain of tan(x) to the interval ( π 2, π 2 ). (Note the end points are NOT included!) () June 6, / 68

20 The Inverse Tangent Function (a.k.a. arctangent function) Definition: For all real numbers x, the inverse tangent of x is denoted by and is defined by the relationship tan 1 (x) or by arctan(x) y = tan 1 (x) x = tan(y) where π 2 < y < π 2. The Domain of the Inverse Tangent is < x <. The Range of the Inverse Tangent is π 2 < y < π 2 inequalities.). (Note the strict () June 6, / 68

21 The Graph of the Arctangent Figure: The domain is all real numbers and the range is π 2 < y < π 2. The graph has two horizontal asymptotes y = π 2 and y = π 2. () June 6, / 68

22 Function/Inverse Function Relationship For all real numbers x ( ) tan tan 1 (x) = x For every x in the interval ( π 2, π 2 ) tan 1 (tan(x)) = x Remark 1:The expression tan 1 (x) is always well defined. Remark 2: If x > π 2 or x < π 2, the expression tan 1 (tan(x)) MAY BE defined, but IS NOT equal to x. () June 6, / 68

23 Some Inverse Tangent Values We can build a table of some inverse tangent values by using our knowledge of the tangent function. x tan(x) π 3 3 π 4 1 π π π 4 1 π 3 3 x tan 1 (x) 3 π 3 1 π π π 3 6 π π 3 () June 6, / 68

24 Conceptual Definition We can think of the inverse tangent function in the following way: tan 1 (x) is the angle between π 2 and π 2 whose tangent is x. () June 6, / 68

25 Examples Evaluate each expression exactly. (a) tan 1 ( 1) (b) tan 1 (0) ( ) 1 (c) tan 1 3 () June 6, / 68

26 Examples Evaluate each expression if possible. If undefined, state a reason. [ ] (a) tan tan 1 (1) (b) [ ] tan tan 1 (52.5) () June 6, / 68

27 ( )] 2π (c) tan [tan 1 7 ( (d) tan [tan 1 5π 6 )] () June 6, / 68

28 Combining Functions Evaluate each expression exactly if possible. If it is undefined, provide a reason. (a) [ ( )] 1 tan arcsin 2 (b) tan [arccos (0)] () June 6, / 68

29 (c) [ ( )] 1 sin tan 1 4 () June 6, / 68

30 (d) ( cos [sin 1 2 )] 3 () June 6, / 68

31 Write the expression purely algebraically (with no trigonometric functions). [ ] (e) tan sin 1 (x) () June 6, / 68

32 Write the expression purely algebraically (with no trigonometric functions). ( )] x (f) cos [tan () June 6, / 68

33 Recap: Inverse Sine, Cosine, and Tangent Function Domain Range sin 1 [ (x) [ 1, 1] π 2, ] π 2 cos 1 (x) [ 1, 1] [0, π] tan 1 (x) (, ) ( π 2, π 2 ) () June 6, / 68

34 Application Example () June 6, / 68

35 () June 6, / 68

36 Three Other Inverse Trigonometric Functions There is disagreement about how to define the ranges of the inverse cotangent, cosecant, and secant functions! The inverse Cotangent function is typically defined for all real numbers x by y = cot 1 (x) x = cot(y) for 0 < y < π. There is less consensus regarding the inverse secant and cosecant functions. () June 6, / 68

37 A Work-Around Use the fact that sec θ = 1 cos θ to find an expression for sec 1 (x). Use your result to compute (an acceptable value for) sec 1 ( 2) () June 6, / 68

38 A Work-Around We can use the following compromise to compute inverse cotangent, secant, and cosecant values 1 : For those values of x for which each side of the equation is defined ( ) 1 cot 1 (x) = tan 1, x csc 1 (x) = sin 1 ( 1 x sec 1 (x) = cos 1 ( 1 x ), ). 1 Keep in mind that there is disagreement about the ranges! () June 6, / 68

39 Section 7.1: Fundamental Identities and Their Uses We recall that the trigonometric functions are largely interrelated. For example, we already know that tan(x) = sin(x) cos(x). An important property of this statement is that it is true for every real number x for which the sides of the equation are defined! In this chapter, we will explore various relationships between the trigonometric functions. () June 6, / 68

40 Expressions and Types of Statements To proceed, let s make a distinction between an expression, a statement, a conditional statement, and an identity. () June 6, / 68

41 Expression -vs- Statement Definition: An expression is a grouping of numbers, symbols, and/or operators that may define a mathematical object or quantity. An expression is a NOUN. Examples: Each of x 2 + 7x, ln(sin θ), 25 7 and sin(x) cos(x) are expressions. () June 6, / 68

42 Expression -vs- Statement Definition: A mathematical statement is an assertion that may be true or false. A statement is a SENTENCE. Examples: Some examples of statements include ( π ) x 2 + 7x = 18, sin(0 ) > 12, cos 2 x = sin(x) If x > 0, and x < 0, then x is a green eyed lion. In a symbolic sentence, the verb is usually included in one of the symbols =,,, >, or <. () June 6, / 68

43 Conditional Statement Definition: A conditional statement is a statement that is true under certain conditions. Typically, it is true when a variable is assigned a certain value/values. But it is false when other values are assigned. For example: x 2 + 7x = 18 is true if x = 2 or if x = 9. If any other value is assigned to x, the statement is false. For example: cos(x) = sin(x) is true if x = π 4. It is also true if x = 5π 4. In fact, it is true for infinitely many different values of x. However, it is NOT ALWAYS true. If x = π 2, then the statement is false. () June 6, / 68

44 Identity Definition: An identity is a mathematical statement that is ALWAYS true. If an identity is stated as an equation, this means that for every value of any variable such that both sides of the equation are defined, both sides of the equation are the same. For example: ( π ) csc 2 x = sec(x) is true for every real number x for which each side is defined. For example: cos( x) = cos(x) is true for every real number x for which each side is defined. () June 6, / 68

45 Identities We Already Know Reciprocal: csc(x) = 1 sin(x), sec(x) = 1 cos(x),, cot(x) = 1 tan(x), sin(x) = 1 csc(x), cos(x) = 1 sec(x),, tan(x) = 1 cot(x), Quotient: tan(x) = sin(x) cos(x), cos(x) cot(x) = sin(x) () June 6, / 68

46 Identities We Already Know Cofunction: ( π ) cos(x) = sin 2 x, ( π ) sin(x) = cos 2 x, ( π ) csc(x) = sec 2 x, ( π ) sec(x) = csc 2 x, ( π ) cot(x) = tan 2 x, ( π ) tan(x) = cot 2 x. Periodicity: sin (x + 2π) = sin(x), cos (x + 2π) = cos(x) csc (x + 2π) = csc(x), sec (x + 2π) = sec(x) tan (x + π) = tan(x), cot (x + π) = cot(x) Symmetry: sin( x) = sin(x), cos( x) = cos(x), tan( x) = tan(x) csc( x) = csc(x), sec( x) = sec(x), cot( x) = cot(x). () June 6, / 68

47 Pythagorean Identities sin 2 (x) + cos 2 (x) = 1 Figure: Triangle in a unit circle. This Pythagorean ID. follows directly from the Pythagorean Theorem. () June 6, / 68

48 Notation To indicate a power n different from -1 of a trigonometric function, it is standard to write sin n (x), or cos n (x). For example tan 3 (x) is read the tangent cubed of x. Note that tan 3 (x) = (tan(x)) 3 We usually write this way for integer powers n. We NEVER write this for the power 1. () June 6, / 68

49 Pythagorean Identities Use sin 2 (x) + cos 2 (x) = 1 along with other identities to show that tan 2 (x) + 1 = sec 2 (x) () June 6, / 68

50 Pythagorean Identities Use sin 2 (x) + cos 2 (x) = 1 along with other identities to show that 1 + cot 2 (x) = csc 2 (x) () June 6, / 68

51 Pythagorean Identities We have the three new identities: sin 2 (x) + cos 2 (x) = 1 tan 2 (x) + 1 = sec 2 (x), and 1 + cot 2 (x) = csc 2 (x) Note the the squares appearing here are NOT OPTIONAL! They are critical to the identities. () June 6, / 68