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MTH 112 Elementary Functions Chapter 6 Trigonometric Identities, Inverse Functions, and Equations

MTH 112 Elementary Functions Chapter 6 Trigonometric Identities, Inverse Functions, and Equations. Section 4 Inverses of the Trigonometric Functions. Inverse of a Function?. Function: f = { (a,b) | aD bR  f(a) = b } D = Domain of the Function R = Range of the Function

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MTH 112 Elementary Functions Chapter 6 Trigonometric Identities, Inverse Functions, and Equations

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  1. MTH 112Elementary FunctionsChapter 6Trigonometric Identities, Inverse Functions, and Equations Section 4 Inverses of the Trigonometric Functions

  2. Inverse of a Function? • Function: f = { (a,b) | aD bR  f(a) = b } • D = Domain of the Function • R = Range of the Function • Inverse: f -1 = { (b,a) | (a,b)f and f -1(b) = a} • Basic relationship: f -1(f(x)) = x and f(f -1(x)) = x • Does the inverse of a function exist? • Only if the function is one-to-one. • That is, if f(a) = f(b) then a = b for all a & b in the domain of f.

  3. Examples • Does the inverse of y = x2 exist? • No, because a2 = b2 does not imply that a = b. • Does the inverse of y = x2 where x  0 exist? • Yes! • Do these two functions have the same range? • Yes! • Note that if the inverse of a function does not exist, sometimes the inverse of the original function with a restricted domain and same range does exist.

  4. a a b b y = x (b,a) (a,b) The Graph of theInverse of a Function • What is the relationship between the graph of a function and the graph of its inverse? • Reflection about the line y = x. Why? • Each ordered pair (a,b) of the function corresponds to an ordered pair (b,a) of the inverse.

  5. Do the inverses of the trig functions exist? NO! Try the following on a calculator. sin-1(sin(1)) sin-1(sin(2)) sin-1(sin(4)) sin-1(sin(6)) sin-1(sin(-1)) sin-1(sin(-2)) sin-1(sin(-4)) sin-1(sin(-6)) Inverses of Trig Functions Note: 1 radian  57° 2 radians  115° 4 radians  229° 6 radians  344°

  6. Do the inverses of the trig functions exist? NO! Try the following on a calculator. sin-1(sin(1)) = 1 sin-1(sin(2)) 1.14 sin-1(sin(4)) -0.86 sin-1(sin(6)) -0.28 sin-1(sin(-1)) = -1 sin-1(sin(-2)) -1.14 sin-1(sin(-4)) -0.86 sin-1(sin(-6)) -0.28 Inverses of Trig Functions Note: 1 radian  57° 2 radians  115° 4 radians  229° 6 radians  344°

  7. Do the inverses of the trig functions exist? NO! Try the following on a calculator. cos-1(cos(1)) cos-1(cos(2)) cos-1(cos(4)) cos-1(cos(6)) cos-1(cos(-1)) cos-1(cos(-2)) cos-1(cos(-4)) cos-1(cos(-6)) Inverses of Trig Functions Note: 1 radian  57° 2 radians  115° 4 radians  229° 6 radians  344°

  8. Do the inverses of the trig functions exist? NO! Try the following on a calculator. cos-1(cos(1)) = 1 cos-1(cos(2)) = 2 cos-1(cos(4)) 2.28 cos-1(cos(6)) 0.28 cos-1(cos(-1)) = 1 cos-1(cos(-2)) = 2 cos-1(cos(-4)) 2.28 cos-1(cos(-6)) 0.28 Inverses of Trig Functions Note: 1 radian  57° 2 radians  115° 4 radians  229° 6 radians  344°

  9. Try the following on a calculator. sin(sin-1(1)) sin(sin-1(2)) sin(sin-1(-1)) sin(sin-1(-2)) sin(sin-1(0.5)) sin(sin-1(-0.5)) Inverses of Trig Functions

  10. Try the following on a calculator. sin(sin-1(1)) = 1 sin(sin-1(2)) = error sin(sin-1(-1)) = -1 sin(sin-1(-2)) = error sin(sin-1(0.5)) = 0.5 sin(sin-1(-0.5)) = -0.5 Try the following on a calculator. cos(cos-1(1)) cos(cos-1(2)) cos(cos-1(-1)) cos(cos-1(-2)) cos(cos-1(0.5)) cos(cos-1(-0.5)) Inverses of Trig Functions

  11. Try the following on a calculator. sin(sin-1(1)) = 1 sin(sin-1(2)) = error sin(sin-1(-1)) = -1 sin(sin-1(-2)) = error sin(sin-1(0.5)) = 0.5 sin(sin-1(-0.5)) = -0.5 Try the following on a calculator. cos(cos-1(1)) = 1 cos(cos-1(2)) = error cos(cos-1(-1)) = -1 cos(cos-1(-2)) = error cos(cos-1(0.5)) = 0.5 cos(cos-1(-0.5)) = -0.5 Inverses of Trig Functions

  12. y = sin x (-3/2, 1) (/2, 1) -2 -  2 (3/2, -1) (-/2, -1) y = sin-1x = arcsin x Begin with y = sin x. Restrict the domain to [-/2, /2]. This gives a 1-1 function with the same range as the original function.

  13. y = sin-1x = arcsin x • Therefore … • y = sin-1x  x = sin y, where y  [-/2, /2] • Domain: [-1, 1] • Range: [-/2, /2]

  14. (-2, 1) (0, 1) (2, 1) -/2 -3/2 /2 3/2 y = cos x (, -1) (-, -1) y = cos-1x = arccos x Begin with y = cos x. Restrict the domain to [0, ]. This gives a 1-1 function with the same range as the original function.

  15. y = cos-1x = arccos x • Therefore … • y = cos-1x  x = cos y, where y  [0, ] • Domain: [-1, 1] • Range: [0, ]

  16. y = tan x (, 0) (2, 0) (-, 0) (-2, 0) (0, 0) -3/2 -/2 /2 3/2 y = tan-1x = arctan x Begin with y = tan x. Restrict the domain to (-/2, /2). This gives a 1-1 function with the same range as the original function.

  17. y = tan-1x = arctan x • Therefore … • y = tan-1x  x = tan y, where y  (-/2, /2) • Domain: (-, ) • Range: (-/2, /2)

  18. Likewise, the other three … • y = sec-1x = arcsec x  x = sec y • Domain: (-, -1]  [1, ) • Range: [0, /2)  (/2, ] • y = csc-1x = arccsc x  x = csc y • Domain: (-, -1]  [1, ) • Range: [-/2, 0)  (0, /2] • y = cot-1x = arccot x  x = cot y • Domain: (-, ) • Range: [-/2, 0)  (0, /2] • Why not (0, )? Because of calculators!

  19. sec-1x, csc-1x & cot-1xOn a Calculator y = sec-1x  x = sec y  x = 1/cos y  cos y = 1/x  y = cos-1(1/x)

  20. sec-1x, csc-1x & cot-1xOn a Calculator • Therefore … • y = sec-1x  y = cos-1(1/x) • y = csc-1x  y = sin-1(1/x) • y = cot-1x  y = tan-1(1/x)

  21. Composition of Trig Functionswith Inverse Trig Functions • Under what conditions does sin-1(sin x) = x ? • The range of Inverse Sine function is [-/2, /2] • Therefore, x [-/2, /2].

  22. Composition of Trig Functionswith Inverse Trig Functions • sin-1(sin x) = x  x [-/2, /2] • cos-1(cos x) = x  x [0, ] • tan-1(tan x) = x  x (-/2, /2) • What about something like .. sin-1(sin 5/6) = ? (5/6 is not in the above interval) = sin-1(1/2) = /6 • It can still be evaluated, it’s just not equal to 5/6.

  23. Composition of Trig Functionswith Inverse Trig Functions • Under what conditions does sin(sin-1x) = x ? • The domain of Inverse Sine function is [-1, 1] • Therefore, x [-1, 1].

  24. Composition of Trig Functionswith Inverse Trig Functions • sin(sin-1x) = x  x [-1, 1] • cos(cos-1x) = x  x [-1, 1] • tan(tan-1x) = x  x (-, ) • What about something like .. sin(sin-1(5/3)) = ? (5/3 is not in the above interval) • It can not be evaluated, because 5/3 is not in the domain of the inverse sine function.

  25. Composition of Trig Functionswith Inverse Trig Functions • What about something like ... sin-1(cos x) • If cos x is a known value, evaluate it and then find the angle whose sine is this value. • If cos x is not a known value, use a calculator. • Example: sin-1(cos(4/3)) = sin-1(-1/2) = -/6

  26. 1 cos-1x x Composition of Trig Functionswith Inverse Trig Functions • What about something like ... sin(cos-1x) • This can always be evaluated without a calculator. • Remember: cos-1x represents an acute angle (if x > 0)! • Draw a triangle where cos-1x is one of the acute angles. • Using the definition, if the adjacent side is x, then the hypotenuse will be 1. • The opposite side will then be Therefore,

  27. x tan-1x 1 Composition of Trig Functionswith Inverse Trig Functions • Another variation ... same approach: csc(tan-1x) • tan-1x represents an acute angle (if x > 0)! • Draw a triangle where tan-1x is one of the acute angles. • Using the definition, if the opposite side is x, then the adjacent side will be 1. • The hypotenuse will then be Therefore,

  28. Composition of Trig Functionswith Inverse Trig Functions Other combinations?

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