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3.1 Inverse Trig Functions

Review of basic inverses:

Def:

Does every function have an inverse?

Graphical properties:

Domain/Range Properties

Def of y = Arcsin x = Sin-1 x

[pic]

Domain: [-1, 1] Range: [-π/2, π/2]

Def of y = Arccos x = Cos-1 x

[pic]

Domain: [-1, 1] Range: [0, π]

Def of y = Arctan x = Tan-1 x

[pic]

Domain: (-∞, ∞) Range: (-π/2, π/2)

Note: Angles are ALWAYS in RADIANS!!!!!!!

Examples: Evaluate the following EXACTLY:

a) Cos-1(1/2)

b) Arcsin(1/2)

c) Tan-1((3)

d) Cos-1(-(3/2)

e) Sin-1(-(3/2)

f) Sin-1((3/2)

g) Tan-1(-1)

h) Arccos(-(2/2)

i) Tan-1(-(3/3)

j) Sin-1((3)

k) cos(π/3)

l) Cos-1(π/3)

Determine if the following statements are True T or False F. Explain WHY.

To do these we MUST KNOW the domain and range of the trig functions

| |Domain |Range |

|[pic] | | |

|[pic] | | |

|[pic] | | |

[pic] only if x in the domain of composition i.e in the domain of f-1 and range of f

[pic] only if x in the domain of composition i.e in the domain of f and range of f-1

a) Arcsin(sin(π/3))= π/3 True / False

because

b) Arcsin(sin(7π/6))= 7π/6 True / False

because

c) cos-1(cos(2π/3))= 2π/3 True / False

because

d) cos-1(cos(-π/3))= -π/3 True / False

because

e) cos(cos-1(-1/2))=-1/2 True / False

because

f) sin(sin-1(π))=π True / False

because

g) tan-1(tan 5π/3) = 5π/3 True / False

because

Calculator problems: You must be in Radian mode!! Always give answer accurate to 4 decimals unless specified otherwise.

cos-1(-1/2)

sin-1(1/5)

cos-1(π/2)

tan-1(-8.2)

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