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Inverse Trig Functions.notebook

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Pre-CalculusChapter 7Inverse Trig Functions


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• Be able to evaluate inverse trigonometric funcons

• Be able to explain the differences between a trig funcon and its inverse

Objective:

• Be able to write equaons for inverses of trigonometric funcons• Graph inverses of trigonometric funcons


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PreCalculus

Graphing Inverses of Trig functions


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Cos1 (x) takes a value x and produces an angle y

and Cos(x) and 1/Cos(x) =sec(x) takes an angle x and produces a number value y

so these functions use two different types of x's therefore their y's will never equal each other

Inverse functions undo the functions so we can get the input value.

Inverse Trig Functions:


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sin(x) = y x = sin‐1(y) or arcsin(y) = xcos(x) = y x = cos‐1(y) or arccos(y) = xtan(x) = y x = tan‐1(y) or arctan(y) = x

Inverse Trig Functions


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Examples

A = cos(b) cos‐1 (A) = b or arccos(A) = b

Tan(θ) = z tan‐1 (z) = θ or arctan(z)= θ

cot(m) = p cot‐1 (p) = m or arccot(p)= m

c = csc(d) csc‐1 (c) = d or arccsc(c) = d


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ex. Find all posive values of x for which cosx = ½ look on your unit circlex = cos‐1 (1/2)

so x = 60, 300, 420….. and π/3, 5 π/3, 7 π/3……so between 0 and 360, we have 60 and 300

ex. Find all posive values of x for which sin x = (1/ √2) between 0 and 360

Sinx = so 45, 135


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Ex. Evaluate each expression. Assume that all angles are in quad 1

cos(arccos (½)) = ½

cos(arccsc(5/3)) = 4/5

sec(arctan (7/13)) = √218/13

On your own: sin(arcsin(13/14) = 13/14

cot(tan‐1(5/10) = 10/5


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Ex. Verify each equaon.arccos + arcsin = arctan 1 + arccot 1

tan‐1 (3/4) + tan‐1 (5/12) = tan‐1 (56/33)

On your own:arcsin (3/5) + arcos (15/17) = arctan (77/36)

tan‐1 1 + cos‐1 = sin‐1 1/2 + sec‐1 √2


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sin(x)Domain (all reals)Range(‐1<y<1)(x, sin(x))(90o, 1)Funconsin(x) = (1/2)The value of sine of an angle x is 1/2

sin‐1(x) or arcsin(x)Domain [‐1,1]Range (all reals)(sin(x), x)(1, 90o)not a funcon (failed vercal line test)x = sin‐1(1/2)x is all the angles whose sine is 1/2, x = 30o, 150,and so on

Sin(x)Domain [‐90,90]Range[‐1,1](x, sin(x))(90o, 1)Funconsin(x) = (1/2)The value of sine of an angle x is 1/2

Sin‐1(x) or Arcsin(x)Domain [‐1,1]Range [‐90,90](sin(x), x)(1, 90o)not a funcon (failed vercal line test)x = sin‐1(1/2)x is only the angles whose sine is 1/2 and

between ‐90o and 90o so only 30o


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cos(x)Domain (all reals)Range(‐1<y<1)(x, cos(x))(90o, 0)Funconcos(x) = (1/2)The value of cosine of an angle x is 1/2

cos‐1(x) or arccos(x)Domain [‐1,1]Range (all reals)(cos(x), x)(0, 90o)not a funcon (failed vercal line test)x = cos‐1(1/2)x is all the angles whose cosine is 1/2, x=60 o,120o, and so on

Cos(x)Domain [0,180]Range[0,1](x, cos(x))(90o, 0)Funconcos(x) = (1/2)The value of cosine of an angle x is 1/2

Cos‐1(x) or Arccos(x)Domain [‐1,1]Range [0,180](cos(x), x)(0, 90o)Funcon! (now it passes vercal line test)x = Cos‐1(1/2)x is only the angles whose cosine is 1/2 and

between 0o and 180o which would be only 60 o


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tan(x) = y

X Y X Y

tan‐1x=y or

arctan(x) = y

Tan‐1x=y or

Arctan(x) = y


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cot(x) = y

X Y X Y

cot‐1x=y or

arccot(x) = y


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Cot(x) = y

X Y X Y

Cot‐1(x) = y or

Arccot(x) = y


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sec(x) = y

X Y X Ysec‐1(x) = y or

arcsec(x) = y


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Sec(x) = y

X Y X Y Sec‐1x = y or Arcsec(x) = y


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csc(x) = y

X Y X Y csc‐1(x) = y or

arccsc(x) = y


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Csc(x) = y

X Y X Y Csc‐1(x) = y or

Arccsc(x) = y


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1) What is the domain of the funcon 2)write the inverse funcon (Switch x and y, solve for y)

Y = Arctan(x) :

Y = Sin(x) ‐ 45 :

Y = Arctan (2x) :


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Determine if each of the following is true or false. If false, give a counterexample.

Make table: x, Inside parenthesis, outside parenthesis; see if last matches first

tan(Tan‐1(x)) = x for all x

Cot‐1(cot(x)) = x for all x

Cos‐1 (x) = 1/Cos(x)

X Tan1(x) tan(Tan1x)

X cot(x) Cot1(cot(x))

X Cos1(x) 1/Cos(x)

1 45 12 63.4 23 71.5 3

45 1 4590 0 90135 1 45


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• True or False: if sin(x) = y then y is a number value between ‐1 and 1• True or False: if arcsin(x) = y then y is an angle without a range• True or False: Arcsin(x)=y is exactly the same as arcsin(x)=y except for

the fact that arcsin(x) has a restricted domain.• True or False: The domain of Tan‐1(x) =y is restricted• True or False: The restricted range of Arccos(x)=y is the restricted domain of Cos(x)=y

Evaluaon:

Practice:Finish pg. 429

Do pg. 452 #2931,35,36, finish Review Packet and worksheet


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pg. 331 16 – 44 evens

Homework:

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